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Page 1: Theorems in Algebra

PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information.PDF generated at: Sun, 16 Oct 2011 18:55:48 UTC

Theorems in Algebra

Page 2: Theorems in Algebra

ContentsArticles

Abel's binomial theorem 1Abel–Ruffini theorem 1Abhyankar's conjecture 4Acyclic model 5Ado's theorem 7Alperin–Brauer–Gorenstein theorem 8Amitsur–Levitzki theorem 9Artin approximation theorem 10Artin–Wedderburn theorem 11Artin–Zorn theorem 12Baer–Suzuki theorem 12Beauville–Laszlo theorem 13Binomial inverse theorem 15Binomial theorem 16Birch's theorem 24Birkhoff's representation theorem 25Boolean prime ideal theorem 29Borel–Weil theorem 32Borel–Weil–Bott theorem 34Brauer's theorem on induced characters 36Brauer's three main theorems 37Brauer–Cartan–Hua theorem 39Brauer–Nesbitt theorem 40Brauer–Siegel theorem 40Brauer–Suzuki theorem 41Brauer–Suzuki–Wall theorem 42Burnside theorem 42Cartan's theorem 43Cartan–Dieudonné theorem 44Cauchy's theorem (group theory) 44Cayley's theorem 46Cayley–Hamilton theorem 48Chevalley–Shephard–Todd theorem 57Chevalley–Warning theorem 59

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Classification of finite simple groups 61Cohn's irreducibility criterion 68Cramer's rule 69Crystallographic restriction theorem 76Descartes' rule of signs 81Dirichlet's unit theorem 83Engel theorem 86Factor theorem 87Feit–Thompson theorem 88Fitting's theorem 91Focal subgroup theorem 91Frobenius determinant theorem 95Frobenius theorem (real division algebras) 96Fundamental lemma (Langlands program) 98Fundamental theorem of algebra 99Fundamental theorem of cyclic groups 107Fundamental theorem of Galois theory 109Fundamental theorem of linear algebra 112Fundamental theorem on homomorphisms 114Gilman–Griess theorem 115Going up and going down 115Goldie's theorem 117Golod–Shafarevich theorem 118Gorenstein–Harada theorem 119Gromov's theorem on groups of polynomial growth 120Grushko theorem 121Haboush's theorem 124Hahn embedding theorem 126Hajós's theorem 127Harish-Chandra isomorphism 128Hasse norm theorem 130Hasse–Arf theorem 130Hilbert's basis theorem 132Hilbert's irreducibility theorem 134Hilbert's Nullstellensatz 135Hilbert's syzygy theorem 137Hilbert's Theorem 90 137Hopkins–Levitzki theorem 138

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Hurwitz's theorem (normed division algebras) 139Isomorphism extension theorem 140Isomorphism theorem 140Jacobson density theorem 145Jordan's theorem (symmetric group) 147Jordan–Schur theorem 148Krull's principal ideal theorem 149Krull–Schmidt theorem 149Künneth theorem 151Kurosh subgroup theorem 154Lagrange's theorem (group theory) 155Lasker–Noether theorem 157Latimer-MacDuffee theorem 160Lattice theorem 160Levitzky's theorem 161Lie's third theorem 162Lie–Kolchin theorem 162Maschke's theorem 164Milnor conjecture 165Mordell–Weil theorem 166Multinomial theorem 167Nielsen–Schreier theorem 170Perron–Frobenius theorem 172Poincaré–Birkhoff–Witt theorem 184Polynomial remainder theorem 187Primitive element theorem 188Quillen–Suslin theorem 190Rational root theorem 191Regev's theorem 193Schreier refinement theorem 194Schur–Zassenhaus theorem 194Serre–Swan theorem 195Skolem–Noether theorem 196Specht's theorem 197Stone's representation theorem for Boolean algebras 199Structure theorem for finitely generated modules over a principal ideal domain 200Subgroup test 204Subring test 205

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Sylow theorems 205Sylvester's determinant theorem 211Sylvester's law of inertia 211Takagi existence theorem 213Three subgroups lemma 214Trichotomy theorem 215Walter theorem 216Wedderburn's little theorem 217Weil conjecture on Tamagawa numbers 218Witt's theorem 219Z* theorem 220Zassenhaus lemma 221ZJ theorem 222

ReferencesArticle Sources and Contributors 223Image Sources, Licenses and Contributors 228

Article LicensesLicense 229

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Abel's binomial theorem 1

Abel's binomial theoremAbel's binomial theorem, named after Niels Henrik Abel, states the following:

Example

m = 2

References• Weisstein, Eric W., "Abel's binomial theorem [1]" from MathWorld.

References[1] http:/ / mathworld. wolfram. com/ AbelsBinomialTheorem. html

Abel–Ruffini theoremIn algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no generalalgebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.[1]

InterpretationThe content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equationsare unsolvable. In fact, the opposite is true: every non-constant polynomial equation in one unknown, with real orcomplex coefficients, has at least one complex number as solution; this is the fundamental theorem of algebra.Although the solutions cannot always be expressed exactly with radicals, they can be computed to any desireddegree of accuracy using numerical methods such as the Newton–Raphson method or Laguerre method, and in thisway they are no different from solutions to polynomial equations of the second, third, or fourth degrees.The theorem only concerns the form that such a solution must take. The theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational constants, and repeatedly forming sums, differences, products, quotients, and radicals (n-th roots, for some integer n) of previously obtained numbers. This clearly excludes the possibility of having any formula that expresses the solutions of an arbitrary equation of degree 5 or higher in terms of its coefficients, using only those operations, or even of having different formulas for different roots or for different classes of polynomials, in such a way as to cover all cases. (In principle one could imagine formulas using irrational numbers as constants, but even if a finite number of those were admitted at the start, not all roots of higher-degree equations could be obtained.) However some polynomial equations, of arbitrarily high degree, are solvable with such operations. Indeed if the roots happen to be rational

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AbelRuffini theorem 2

numbers, they can trivially be expressed as constants. The simplest nontrivial example is the equation , whosesolutions are

Here the expression , which appears to involve the use of the exponential function, in fact just gives thedifferent possible values of (the n-th roots of unity), so it involves only extraction of radicals.

Lower-degree polynomialsThe solutions of any second-degree polynomial equation can be expressed in terms of addition, subtraction,multiplication, division, and square roots, using the familiar quadratic formula: The roots of the following equationare shown below:

Analogous formulas for third- and fourth-degree equations, using cube roots and fourth roots, had been known sincethe 16th century.

Quintics and higherThe Abel–Ruffini theorem says that there are some fifth-degree equations whose solution cannot be so expressed.The equation is an example. (See Bring radical.) Some other fifth degree equations can be solvedby radicals, for example , which factorizes to

. The precise criterion that distinguishes between those equationsthat can be solved by radicals and those that cannot was given by Évariste Galois and is now part of Galois theory: apolynomial equation can be solved by radicals if and only if its Galois group (over the rational numbers, or moregenerally over the base field of admitted constants) is a solvable group.Today, in the modern algebraic context, we say that second, third and fourth degree polynomial equations canalways be solved by radicals because the symmetric groups S2, S3 and S4 are solvable groups, whereas Sn is notsolvable for n ≥ 5. This is so because for a polynomial of degree n with indeterminate coefficients (i.e., given bysymbolic parameters), the Galois group is the full symmetric group Sn (this is what is called the "general equation ofthe n-th degree"). This remains true if the coefficients are concrete but algebraically independent values over thebase field.

ProofThe following proof is based on Galois theory. Historically, Ruffini and Abel's proofs precede Galois theory.One of the fundamental theorems of Galois theory states that an equation is solvable in radicals if and only if it has asolvable Galois group, so the proof of the Abel–Ruffini theorem comes down to computing the Galois group of thegeneral polynomial of the fifth degree.

Let be a real number transcendental over the field of rational numbers , and let be a real numbertranscendental over , and so on to which is transcendental over . These numbers arecalled independent transcendental elements over Q. Let and let

Multiplying out yields the elementary symmetric functions of the :

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AbelRuffini theorem 3

and so on up to

The coefficient of in is thus . Because our independent transcendentals act asindeterminates over , every permutation in the symmetric group on 5 letters induces an automorphism on that leaves fixed and permutes the elements . Since an arbitrary rearrangement of the roots of theproduct form still produces the same polynomial, e.g.:

is still the same polynomial as

the automorphisms also leave fixed, so they are elements of the Galois group . Now, sinceit must be that , as there could possibly be automorphisms there that are not in .

However, since the splitting field of a quintic polynomial has at most 5! elements, , and somust be isomorphic to . Generalizing this argument shows that the Galois group of every general

polynomial of degree is isomorphic to .And what of ? The only composition series of is (where is the alternating group onfive letters, also known as the icosahedral group). However, the quotient group (isomorphic to itself)is not an abelian group, and so is not solvable, so it must be that the general polynomial of the fifth degree has nosolution in radicals. Since the first nontrivial normal subgroup of the symmetric group on n letters is always thealternating group on n letters, and since the alternating groups on n letters for are always simple andnon-abelian, and hence not solvable, it also says that the general polynomials of all degrees higher than the fifth alsohave no solution in radicals.Note that the above construction of the Galois group for a fifth degree polynomial only applies to the generalpolynomial, specific polynomials of the fifth degree may have different Galois groups with quite different properties,e.g. has a splitting field generated by a primitive 5th root of unity, and hence its Galois group is abelian andthe equation itself solvable by radicals. However, since the result is on the general polynomial, it does say that ageneral "quintic formula" for the roots of a quintic using only a finite combination of the arithmetic operations andradicals in terms of the coefficients is impossible. Q.E.D.

HistoryAround 1770, Joseph Louis Lagrange began the groundwork that unified the many different tricks that had been usedup to that point to solve equations, relating them to the theory of groups of permutations, in the form of Lagrangeresolvents. This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutionsfor equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provideconclusive proof. The theorem, however, was first nearly proved by Paolo Ruffini in 1799, but his proof was mostlyignored. He had several times tried to send it to different mathematicians to get it acknowledged, amongst them,French mathematician Augustin-Louis Cauchy, but it was never acknowledged, possibly because the proof wasspanning 500 pages. The proof also, as was discovered later, contained an error. Ruffini assumed that a solutionwould necessarily be a function of the radicals (in modern terms, he failed to prove that the splitting field is one ofthe fields in the tower of radicals which corresponds to a solution expressed in radicals). While Cauchy felt that theassumption was minor, most historians believe that the proof was not complete until Abel proved this assumption.The theorem is thus generally credited to Niels Henrik Abel, who published a proof that required just six pages in1824.[2]

Insights into these issues were also gained using Galois theory pioneered by Évariste Galois. In 1885, John StuartGlashan, George Paxton Young, and Carl Runge provided a proof using this theory.

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AbelRuffini theorem 4

Notes[1] Jacobson (2009), p. 211.[2] du Sautoy, Marcus. "January: Impossibilities". Symmetry: A Journey into the Patterns of Nature. ISBN 978-0060789411.

References• Edgar Dehn. Algebraic Equations: An Introduction to the Theories of Lagrange and Galois. Columbia University

Press, 1930. ISBN 0-486-43900-3.• Jacobson, Nathan (2009), Basic algebra, 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1• John B. Fraleigh. A First Course in Abstract Algebra. Fifth Edition. Addison-Wesley, 1994. ISBN

0-201-59291-6.• Ian Stewart. Galois Theory. Chapman and Hall, 1973. ISBN 0-412-10800-3.• Abel's Impossibility Theorem at Everything2 (http:/ / www. everything2. net/ title/ Abel%27s+ Impossibility+

Theorem)

External links• MÉMOIRE SUR LES ÉQUATIONS'ALGÉBRIQUES, OU L'ON DÉMONTRE. L'IMPOSSIBILITÉ DE LA

RÉSOLUTION DE L'ÉQUATION GÉNÉRALE. DU CINQUIÈME DEGRÉ (http:/ / www. abelprisen. no/verker/ oeuvres_1881_del1/ oeuvres_completes_de_abel_nouv_ed_1_kap03_opt. pdf)PDF - the first proof on1824 in French

• Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrièmedegré (http:/ / www. abelprisen. no/ verker/ oeuvres_1839/ oeuvres_completes_de_abel_1_kap02_opt. pdf)PDF -the second proof on 1826 in French

Abhyankar's conjectureIn abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups offunction fields of characteristic p.[1] This problem was solved in 1994 by work of Michel Raynaud and DavidHarbater.[2] [3]

The problem involves a finite group G, a prime number p, and a nonsingular integral algebraic curve C defined overan algebraically closed field K of characteristic p.The question addresses the existence of Galois extensions L of K(C), with G as Galois group, and with restrictedramification. From a geometric point of view L corresponds to another curve C′, and a morphism

π : C′ → C.Ramification geometrically, and by analogy with the case of Riemann surfaces, consists of a finite set S of points xon C, such that π restricted to the complement of S in C is an étale morphism. In Abhyankar's conjecture, S is fixed,and the question is what G can be. This is therefore a special type of inverse Galois problem.The subgroup p(G) is defined to be the subgroup generated by all the Sylow subgroups of G for the prime number p.This is a normal subgroup, and the parameter n is defined as the minimum number of generators of

G/p(G).Then for the case of C the projective line over K, the conjecture states that G can be realised as a Galois group of L,unramified outside S containing s + 1 points, if and only if

n ≤ s.This was proved by Raynaud.

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Abhyankar's conjecture 5

For the general case, proved by Harbater, let g be the genus of C. Then G can be realised if and only ifn ≤ s + 2 g.

References[1] Abhyankar, Shreeram (1957), "Coverings of Algebraic Curves", American Journal of Mathematics 79 (4): 825–856, doi:10.2307/2372438.[2] Raynaud, Michel (1994), "Revêtements de la droite affine en caractéristique p > 0", Inventiones Mathematicae 116 (1): 425–462,

doi:10.1007/BF01231568.[3] Harbater, David (1994), "Abhyankar's conjecture on Galois groups over curves", Inventiones Mathematicae 117 (1): 1–25,

doi:10.1007/BF01232232.

External links• Weisstein, Eric W., " Abhyankar's conjecture (http:/ / mathworld. wolfram. com/ AbhyankarsConjecture. html)"

from MathWorld.• A layman's perspective of Abhyankar's conjecture (http:/ / www. math. purdue. edu/ about/ purview/ spring95/

conjecture. html) from Purdue University

Acyclic modelIn algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that twohomology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and SaundersMacLane. They discovered that, when topologists were writing proofs to establish equivalence of various homologytheories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem togeneralize this process.It can be used to prove the Eilenberg–Zilber theorem.

Statement of the theoremLet be an arbitrary category and be the category of chain complexes of -modules. Let

be covariant functors such that:• for .• There are for such that has a basis in , so is a free functor.• is - and -acyclic at these models, which means that for all and all

.Then the following assertions hold:

• Every natural transformation is induced by a natural chain map .• If are natural transformations, are natural chain maps as before and

for all models , then there is a natural chain homotopy between and .• In particular the chain map is unique up to natural chain homotopy.[1]

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Acyclic model 6

Generalizations

Projective and acyclic complexesWhat is above is one of the earliest versions of the theorem. Another version is the one that says that if is acomplex of projectives in an abelian category and is an acyclic complex in that category, then any map

extends to a chain map , unique up to homotopy.

This specializes almost to the above theorem if one uses the functor category as the abelian category. Freefunctors are projective objects in that category. The morphisms in the functor category are natural transformations,so the constructed chain maps and homotopies are all natural. The difference is that in the above version, beingacyclic is a stronger assumption than being acyclic only at certain objects.On the other hand, the above version almost implies this version by letting a category with only one object. Thenthe free functor is basically just free (and hence projective) module. being acyclic at the models (there is onlyone) means nothing else than that the complex is acyclic.

Acyclic classes

Then there is the grand theorem that unifies them all. Let be an abelian category (for example or ). A class of chain complexes over will be called an acyclic class provided:• The 0 complex is in .• The complex belongs to if and only if the suspension of does.• If the complexes and are homotopic and , then .• Every complex in is acyclic.• If is a double complex, all of whose rows are in , then the total complex of belongs to .There are three natural examples of acyclic classes, although doubtless others exist. The first is that of homotopycontractible complexes. The second is that of acyclic complexes. In functor categories (e.g. the category of allfunctors from topological spaces to abelian groups), there is a class of complexes that are contractible on each object,but where the contractions might not be given by natural transformations. Another example is again in functorcategories but this time the complexes are acyclic only at certain objects.Let denote the class of chain maps between complexes whose mapping cone belongs to . Although does notnecessarily have a calculus of either right or left fractions, it has weaker properties of having homotopy classes ofboth left and right fractions that permit forming the class gotten by inverting the arrows in .Let be an augmented endofunctor on , meaning there is given a natural transformation (theidentity functor on ). We say that the chain complex is -presentable if for each , the chain complex

belongs to . The boundary operator is given by

.

We say that the chain complex functor is -acyclic if the augmented chain complex belongs to .Theorem. Let be an acyclic class and the corresponding class of arrows in the category of chain complexes.Suppose that is -presentable and is -acyclic. Then any natural transformation

extends, in the category to a natural transformation of chain functorsand this is unique in up to chain homotopies. If we suppose, in addition, that is

-presentable, that is -acyclic, and that is an isomorphism, then is homotopy equivalence.

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Acyclic model 7

ExampleHere is an example of this last theorem in action. Let be the category of triangulable spaces and be thecategory of abelian group valued functors on . Let be the singular chain complex functor and be thesimplicial chain complex functor. Let be the functor that assigns to each space the space

. Here, is the -simplex and this functor assigns to the sum of as many copies of

each -simplex as there are maps . Then let be defined by . There is an obviousaugmentation and this induces one on . It can be shown that both and are both -presentable and -acyclic (the proof that is not entirely straigtforward and uses a detour through simplicialsubdivision, which can also be handled using the above theorem). The class is the class of homologyequivalences. It is rather obvious that and so we conclude that singular and simplicialhomology are isomorphic on .There are many other examples in both algebra and topology, some of which are described in M. Barr, AcyclicModels. AMS, 2002.

References[1] Dold, Albrecht (1980), Lectures on Algebraic Topology, A Series of Comprehensive Studies in Mathematics, 200 (2nd ed.), Berlin, New

York: Springer-Verlag, ISBN 3-540-10369-4

Ado's theoremIn abstract algebra, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristiczero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theoremstates that L has a linear representation ρ over K, on a finite-dimensional vector space V, that is a faithfulrepresentation, making L isomorphic to a subalgebra of the endomorphisms of V.While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeperresult. Applied to the real Lie algebra of a Lie group G, it does not imply that G has a faithful linear representation(which is not true in general), but rather that G always has a linear representation that is a local isomorphism with alinear group. It was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of NikolaiChebotaryov.The restriction on the characteristic was removed later, by Iwasawa and Harish-Chandra.

References• I. D. Ado, Note on the representation of finite continuous groups by means of linear substitutions, Izv. Fiz.-Mat.

Obsch. (Kazan'), 7 (1935) pp. 1–43 (Russian language)• Ado, I. D. (1947), "The representation of Lie algebras by matrices" [1] (in Russian), Akademiya Nauk SSSR i

Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 2 (6): 159–173, ISSN 0042-1316,MR0027753 translation in Ado, I. D. (1949), "The representation of Lie algebras by matrices", AmericanMathematical Society Translations 1949 (2): 21, ISSN 0065-9290, MR0030946

• Iwasawa, Kenkichi (1948), "On the representation of Lie algebras", Jap. J. Math. 19: 405–426, MR0032613• Harish-Chandra (1949), "Faithful representations of Lie algebras", Annals of Mathematics. Second Series 50:

68–76, ISSN 0003-486X, JSTOR 1969352, MR0028829• Nathan Jacobson, Lie Algebras, pp. 202–203

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Ado's theorem 8

References[1] http:/ / mi. mathnet. ru/ eng/ umn/ v2/ i6/ p159

Alperin–Brauer–Gorenstein theoremIn mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedralor wreathed[1] Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groupsor projective special unitary groups over a finite fields of odd order, depending on a certain congruence, or to theMathieu group . Alperin, Brauer & Gorenstein (1970) proved this in the course of 261 pages. The subdivisionby 2-fusion is sketched there, given as an exercise in Gorenstein (1968, Ch. 7), and presented in some detail in Kwonet al. (1980).

Notes[1] A 2-group is wreathed if it is a nonabelian semidirect product of a maximal subgroup that is a direct product of two cyclic groups of the same

order, that is, if it is the wreath product of a cyclic 2-group with the symmetric group on 2 points.

References• Alperin, J. L.; Brauer, R.; Gorenstein, D. (1970), "Finite groups with quasi-dihedral and wreathed Sylow

2-subgroups.", Transactions of the American Mathematical Society (American Mathematical Society) 151 (1):1–261, doi:10.2307/1995627, ISSN 0002-9947, JSTOR 1995627, MR0284499

• Gorenstein, D. (1968), Finite groups, Harper & Row Publishers, MR0231903• Kwon, T.; Lee, K.; Cho, I.; Park, S. (1980), "On finite groups with quasidihedral Sylow 2-groups" (http:/ / kms.

or. kr/ home/ journal/ include/ downloadPdfJournal.asp?articleuid={71EE4232-6997-4030-8CA7-85CDBCB5A2CC}), Journal of the Korean Mathematical Society17 (1): 91–97, ISSN 0304-9914, MR593804

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AmitsurLevitzki theorem 9

Amitsur–Levitzki theoremIn algebra, the Amitsur–Levitzki theorem states that the algebra of n by n matrices satisfies a certain identity ofdegree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are PI rings such that thesmallest identity they satisfy has degree exactly 2n.

StatementIf A1,...,A2n are n by n matrices then

where the sum is over all (2n)! elements of the symmetric group S2n. (This polynomial is called the standardpolynomial of degree 2n.)

ProofsAmitsur and Levitsky (1950) gave the first proof.Kostant (1958) deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitivecohomology of Lie algebras.Swan (1963) and Swan (1969) gave a simple combinatorial proof as follows. By linearity it is enough to prove thetheorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as adirected edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges.The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from Ato B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges takenin order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the numberof edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem.Razmyslov (1974) gave a proof related to the Cayley–Hamilton theorem.Rosset (1976) gave a short proof using the exterior algebra of a vector space of dimension 2n.

References• Amitsur, A. S.; Levitzki, Jakob (1950), "Minimal identities for algebras" [1], Proceedings of the American

Mathematical Society 1: 449–463, ISSN 0002-9939, JSTOR 2032312, MR0036751• Amitsur, A. S.; Levitzki, Jakob (1951), "Remarks on Minimal identities for algebras" [2], Proceedings of the

American Mathematical Society 2: 320–327, ISSN 0002-9939, JSTOR 2032509, MR?• Formanek, E. (2001), "Amitsur–Levitzki theorem" [3], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,

Springer, ISBN 978-1556080104• Kostant, Bertram (1958), "A theorem of Frobenius, a theorem of Amitsur–Levitski and cohomology theory", J.

Math. Mech. 7: 237–264, doi:10.1512/iumj.1958.7.07019, MR0092755• Razmyslov, Ju. P. (1974), "Identities with trace in full matrix algebras over a field of characteristic zero",

Mathematics of the USSR-Izvestiya 8 (4): 727, doi:10.1070/IM1974v008n04ABEH002126, ISSN 0373-2436,MR0506414

• Rosset, Shmuel (1976), "A new proof of the Amitsur–Levitski identity", Israel Journal of Mathematics 23 (2):187–188, doi:10.1007/BF02756797, ISSN 0021-2172, MR0401804

• Swan, Richard G. (1963), "An application of graph theory to algebra" [4], Proceedings of the AmericanMathematical Society 14: 367–373, ISSN 0002-9939, JSTOR 2033801, MR0149468

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AmitsurLevitzki theorem 10

• Swan, Richard G. (1969), "Correction to "An application of graph theory to algebra"" [5], Proceedings of theAmerican Mathematical Society 21: 379–380, ISSN 0002-9939, JSTOR ?, MR0255439

References[1] http:/ / www. ams. org/ journals/ proc/ 1950-001-04/ S0002-9939-1950-0036751-9/ S0002-9939-1950-0036751-9. pdf[2] http:/ / www. ams. org/ journals/ proc/ 1951-002-02/ S0002-9939-1951-0040285-6/ S0002-9939-1951-0040285-6. pdf[3] http:/ / eom. springer. de/ a/ a110570. htm[4] http:/ / www. ams. org/ journals/ proc/ 1963-014-03/ S0002-9939-1963-0149468-6/ S0002-9939-1963-0149468-6. pdf[5] http:/ / www. ams. org/ journals/ proc/ 1969-021-02/ S0002-9939-1969-0255439-7/ S0002-9939-1969-0255439-7. pdf

Artin approximation theoremIn mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theorywhich implies that formal power series with coefficients in a field k are well-approximated by the algebraic functionson k.

Statement of the theoremLet

x = x1, …, xndenote a collection of n indeterminates,k[[x]] the ring of formal power series with indeterminates x over a field k, and

y = y1, …, yma different set of indeterminates. Let

f(x, y) = 0be a system of polynomial equations in k[x, y], and c a positive integer. Then given a formal power series solutionŷ(x) ∈ k[[x]] there is an algebraic solution y(x) consisting of algebraic functions such that

ŷ(x) ≡ y(x) mod (x)c.

DiscussionGiven any desired positive integer c, this theorem shows that one can find an algebraic solution approximating aformal power series solution up to the degree specified by c. This leads to theorems that deduce the existence ofcertain formal moduli spaces of deformations as schemes.

References• Artin, Michael. Algebraic Spaces. Yale University Press, 1971.

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ArtinWedderburn theorem 11

Artin–Wedderburn theoremIn abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theoremstates that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings overdivision rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. Inparticular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, whereboth n and D are uniquely determined.As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over adivision ring (a simple algebra) is a matrix ring. This is Joseph Wedderburn's original result. Emil Artin latergeneralized it to the case of Artinian rings.Note that if R is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. Forexample, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.The Artin–Wedderburn theorem reduces classifying simple rings over a division ring to classifying division ringsthat contain a given division ring. This in turn can be simplified: The center of D must be a field K. Therefore R is aK-algebra, and itself has K as its center. A finite-dimensional simple algebra R is thus a central simple algebra overK. Thus the Artin–Wedderburn theorem reduces the problem of classifying finite-dimensional central simplealgebras to the problem of classifying division rings with given center.

ExamplesLet R be the field of real numbers, C be the field of complex numbers, and H the quaternions.• Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple

algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.• Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple

algebra over C must be a matrix ring over C.• Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.• Every commutative semisimple ring must be a finite direct product of fields.[1]

References[1] This is clear since matrix rings larger than 1×1 are never commutative.

• P. M. Cohn (2003) Basic Algebra: Groups, Rings, and Fields, pages 137–9.• J.H.M. Wedderburn (1908). "On Hypercomplex Numbers". Proceedings of the London Mathematical Society 6:

77–118. doi:10.1112/plms/s2-6.1.77.

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ArtinZorn theorem 12

Artin–Zorn theoremIn mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternativedivision ring is necessarily a finite field. It was first published by Zorn, but in his publication Zorn credited it toArtin.[1] [2] The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finiteassociative division rings are fields. As a geometric consequence, every finite Moufang plane is the classicalprojective plane over a finite field.[3] [4]

References[1] Zorn, M. (1930), "Theorie der alternativen Ringe", Abh. Math. Sem. Hamburg 8: 123–147.[2] Lüneburg, Heinz (2001), "On the early history of Galois fields", in Jungnickel, Dieter; Niederreiter, Harald, Finite fields and applications:

proceedings of the Fifth International Conference on Finite Fields and Applications Fq5, held at the University of Augsburg, Germany,August 2–6, 1999, Springer-Verlag, pp. 341–355, ISBN 9783540411093, MR1849100.

[3] Shult, Ernest (2011), Points and Lines: Characterizing the Classical Geometries, Universitext, Springer-Verlag, p. 123,ISBN 9783642156267.

[4] McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Springer-Verlag, p. 34, ISBN 9780387954479.

Baer–Suzuki theoremIn mathematical finite group theory, the Baer–Suzuki theorem, proved by Baer (1957) and Suzuki (1965), statesthat if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements ofthe conjugacy class C are contained in a nilpotent subgroup. Alperin & Lyons (1971) gave a short elementary proof.

References• Alperin, J. L.; Lyons, Richard (1971), "On conjugacy classes of p-elements", Journal of Algebra 19: 536–537,

ISSN 0021-8693, MR0289622• Baer, Reinhold (1957), "Engelsche elemente Noetherscher Gruppen", Mathematische Annalen 133: 256–270,

doi:10.1007/BF02547953, ISSN 0025-5831, MR0086815• Gorenstein, D. (1980), Finite groups [1] (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6,

MR569209• Suzuki, Michio (1965), "Finite groups in which the centralizer of any element of order 2 is 2-closed", Annals of

Mathematics. Second Series 82: 191–212, ISSN 0003-486X, JSTOR 1970569, MR0183773

References[1] http:/ / www. ams. org/ bookstore-getitem/ item=CHEL-301-H

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BeauvilleLaszlo theorem 13

Beauville–Laszlo theoremIn mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry thatallows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was provedby Arnaud Beauville and Yves Laszlo (1995).

The theoremAlthough it has implications in algebraic geometry, the theorem is a local result and is stated in its most primitiveform for commutative rings. If A is a ring and f is a nonzero element of A, then we can form two derived rings: thelocalization at f, Af, and the completion at Af, Â; both are A-algebras. In the following we assume that f is a non-zerodivisor. Geometrically, A is viewed as a scheme X = Spec A and f as a divisor (f) on Spec A; then Af is itscomplement Df = Spec Af, the principal open set determined by f, while  is an "infinitesimal neighborhood" D =Spec  of (f). The intersection of Df and Spec  is a "punctured infinitesimal neighborhood" D0 about (f), equal toSpec  ⊗A Af = Spec Âf.Suppose now that we have an A-module M; geometrically, M is a sheaf on Spec A, and we can restrict it to both theprincipal open set Df and the infinitesimal neighborhood Spec Â, yielding an Af-module F and an Â-module G.Algebraically,

(Despite the notational temptation to write , meaning the completion of the A-module M at the ideal Af,unless A is noetherian and M is finitely-generated, the two are not in fact equal. This phenomenon is the main reasonthat the theorem bears the names of Beauville and Laszlo; in the noetherian, finitely-generated case, it is, as noted bythe authors, a special case of Grothendieck's faithfully flat descent.) F and G can both be further restricted to thepunctured neighborhood D0, and since both restrictions are ultimately derived from M, they are isomorphic: we havean isomorphism

Now consider the converse situation: we have a ring A and an element f, and two modules: an Af-module F and anÂ-module G, together with an isomorphism φ as above. Geometrically, we are given a scheme X and both an openset Df and a "small" neighborhood D of its closed complement (f); on Df and D we are given two sheaves whichagree on the intersection D0 = Df ∩ D. If D were an open set in the Zariski topology we could glue the sheaves; thecontent of the Beauville–Laszlo theorem is that, under one technical assumption on f, the same is true for theinfinitesimal neighborhood D as well.Theorem: Given A, f, F, G, and φ as above, if G has no f-torsion, then there exist an A-module M and isomorphisms

consistent with the isomorphism φ: φ is equal to the composition

The technical condition that G has no f-torsion is referred to by the authors as "f-regularity". In fact, one can state astronger version of this theorem. Let M(A) be the category of A-modules (whose morphisms are A-modulehomomorphisms) and let Mf(A) be the full subcategory of f-regular modules. In this notation, we obtain acommutative diagram of categories (note Mf(Af) = M(Af)):

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BeauvilleLaszlo theorem 14

in which the arrows are the base-change maps; for example, the top horizontal arrow acts on objects by M → M ⊗AÂ.Theorem: The above diagram is a cartesian diagram of categories.

Global versionIn geometric language, the Beauville–Laszlo theorem allows one to glue sheaves on an affine scheme over aninfinitesimal neighborhood of a point. Since sheaves have a "local character" and since any scheme is locally affine,the theorem admits a global statement of the same nature. The version of this statement that the authors foundnoteworthy concerns vector bundles:Theorem: Let X be an algebraic curve over a field k, x a k-rational smooth point on X with infinitesimalneighborhood D = Spec k[[t]], R a k-algebra, and r a positive integer. Then the category Vectr(XR) of rank-r vectorbundles on the curve XR = X ×Spec k Spec R fits into a cartesian diagram:

This entails a corollary stated in the paper:Corollary: With the same setup, denote by Triv(XR) the set of triples (E, τ, σ), where E is a vector bundle on XR, τ isa trivialization of E over (X \ x)R (i.e., an isomorphism with the trivial bundle O(X - x)R), and σ a trivialization overDR. Then the maps in the above diagram furnish a bijection between Triv(XR) and GLr(R((t))) (where R((t)) is theformal Laurent series ring).The corollary follows from the theorem in that the triple is associated with the unique matrix which, viewed as a"transition function" over D0

R between the trivial bundles over (X \ x)R and over DR, allows gluing them to form E,with the natural trivializations of the glued bundle then being identified with σ and τ. The importance of thiscorollary is that it shows that the affine Grassmannian may be formed either from the data of bundles over aninfinitesimal disk, or bundles on an entire algebraic curve.

References• Beauville, Arnaud; Laszlo, Yves (1995), "Un lemme de descente" [1], Comptes Rendus de l'Académie des

Sciences Série I. Mathématique 320 (3): 335–340, ISSN 0764-4442, retrieved 2008-04-08

References[1] http:/ / math1. unice. fr/ ~beauvill/ pubs/ descente. pdf

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Binomial inverse theorem 15

Binomial inverse theoremIn mathematics, the Binomial Inverse Theorem is useful for expressing matrix inverses in different ways.If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then

provided A and B + BVA−1UB are nonsingular. Note that if B is invertible, the two B terms flanking the quantityinverse in the right-hand side can be replaced with (B−1)−1, which results in

This is the matrix inversion lemma, which can also be derived using matrix blockwise inversion.

VerificationFirst notice that

Now multiply the matrix we wish to invert by its alleged inverse

which verifies that it is the inverse.

So we get that—if A−1 and exist, then exists and is given by the

theorem above.[1]

Special casesIf p = q and U = V = Ip is the identity matrix, then

Remembering the identity

we can also express the previous equation in the simpler form as

If B = Iq is the identity matrix and q = 1, then U is a column vector, written u, and V is a row vector, written vT.Then the theorem implies

This is useful if one has a matrix with a known inverse A−1 and one needs to invert matrices of the form A+uvT

quickly.If we set A = Ip and B = Iq, we get

In particular, if q = 1, then

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Binomial inverse theorem 16

References[1] Gilbert Strang (2003). Introduction to Linear Algebra (3rd edition ed.). Wellesley-Cambridge Press: Wellesley, MA. ISBN 0-9614088-98.

Binomial theorem

The binomial coefficients appear as the entries ofPascal's triangle.

In elementary algebra, the binomial theorem describes thealgebraic expansion of powers of a binomial. According to thetheorem, it is possible to expand the power (x + y)n into a suminvolving terms of the form axbyc, where the exponents b and c arenonnegative integers with b + c = n, and the coefficient a of eachterm is a specific positive integer depending on n and b. When anexponent is zero, the corresponding power is usually omitted fromthe term. For example,

The coefficient a in the term of xbyc is known as the binomial coefficient or (the two have the same value).These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise incombinatorics, where gives the number of different combinations of b elements that can be chosen from ann-element set.

HistoryThis formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, whodescribed them in the 17th century, but they were known to many mathematicians who preceded him. The 4thcentury B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2[1] [2] asdid the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and theso-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persianmathematician Al-Karaji,[3] , in the 11th century to Persian poet and mathematician Omar Khayyam[4] , and in the13th century to Chinese mathematician Yang Hui, who all derived similar results.[5] Al-Karaji also provided amathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.[3]

Statement of the theoremAccording to the theorem, it is possible to expand any power of x + y into a sum of the form

where each is a specific positive integer known as binomial coefficient. This formula is also referred to as theBinomial Formula or the Binomial Identity. Using summation notation, it can be written as

The final expression follows from the previous one by the symmetry of x and y in the first expression, and bycomparison it follows that the sequence of binomial coefficients in the formula is symmetrical.

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Binomial theorem 17

A variant of the binomial formula is obtained by substituting 1 for x and x for y, so that it involves only a singlevariable. In this form, the formula reads

or equivalently

Examples

Pascal's triangle

The most basic example of the binomialtheorem is the formula for the square ofx + y:

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the third row of Pascal's triangle. Thecoefficients of higher powers of x + y correspond to later rows of the triangle:

Notice that

1. the powers of x go down until it reaches 0 (none of x),starting value is n (the n in 2. the powers of y go up from 0 (none of y)until it reaches n (also the n in 3. the nth row of the Pascal's Triangle will be the coefficients of the expanded binomial.(Note that the top is row 0.)The binomial theorem can be applied to the powers of any binomial. For example,

For a binomial involving subtraction, the theorem can be applied as long as the opposite of the second term is used.This has the effect of changing the sign of every other term in the expansion:

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Binomial theorem 18

Geometrical explanation

For positive values of a and b, the binomial theorem with n = 2is the geometrically evident fact that a square of side a + b canbe cut into a square of side a, a square of side b, and tworectangles with sides a and b. With n = 3, the theorem statesthat a cube of side a + b can be cut into a cube of side a, a cubeof side b, three a×a×b rectangular boxes, and three a×b×brectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative [6] if one sets andinterpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the

volume of an n-dimensional hypercube, where the coefficient of the linear term (in ) is the area of the n faces, each of dimension

Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higherorder terms – and higher – become negligible, and yields the formula interpreted as

"the infinitesimal change in volume of an n-cube as side length varies is the area of n of its -dimensional faces".

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtainsCavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula fordetails.[6]

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Binomial theorem 19

The binomial coefficientsThe coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written

, and pronounced “n choose k”.

FormulasThe coefficient of xn−kyk is given by the formula

,

which is defined in terms of the factorial function n!. Equivalently, this formula can be written

with k factors in both the numerator and denominator of the fraction. Note that, although this formula involves afraction, the binomial coefficient is actually an integer.

Combinatorial interpretation

The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set.This is related to binomials for the following reason: if we write (x + y)n as a product

then, according to the distributive law, there will be one term in the expansion for each choice of either x or y fromeach of the binomials of the product. For example, there will only be one term xn, corresponding to choosing 'x fromeach binomial. However, there will be several terms of the form xn−2y2, one for each way of choosing exactly twobinomials to contribute a y. Therefore, after combining like terms, the coefficient of xn−2y2 will be equal to thenumber of ways to choose exactly 2 elements from an n-element set.

Proofs

Combinatorial proof

Example

The coefficient of xy2 in

equals because there are three x,y strings of length 3 with exactly two y's, namely,

corresponding to the three 2-element subsets of { 1, 2, 3 }, namely,

where each subset specifies the positions of the y in a corresponding string.

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Binomial theorem 20

General case

Expanding (x + y)n yields the sum of the 2 n products of the form e1e2 ... e n where each e i is x or y. Rearrangingfactors shows that each product equals xn−kyk for some k between 0 and n. For a given k, the following are provedequal in succession:• the number of copies of xn − kyk in the expansion• the number of n-character x,y strings having y in exactly k positions• the number of k-element subsets of { 1, 2, ..., n}

• (this is either by definition, or by a short combinatorial argument if one is defining as

).This proves the binomial theorem.

Inductive proofInduction yields another proof of the binomial theorem (1). When n = 0, both sides equal 1, since x0 = 1 for all x and

. Now suppose that (1) holds for a given n; we will prove it for n + 1. For j, k ≥ 0, let [ƒ(x, y)] jk denote

the coefficient of xjyk in the polynomial ƒ(x, y). By the inductive hypothesis, (x + y)n is a polynomial in x and y such

that [(x + y)n] jk is if j + k = n, and 0 otherwise. The identity

shows that (x + y)n+1 also is a polynomial in x and y, and

If j + k = n + 1, then (j − 1) + k = n and j + (k − 1) = n, so the right hand side is

by Pascal's identity. On the other hand, if j +k ≠ n + 1, then (j – 1) + k ≠ n and j +(k – 1) ≠ n, so we get 0 + 0 = 0.Thus

which is the inductive hypothesis with n + 1 substituted for n and so completes the inductive step.

Generalisations

Newton's generalised binomial theoremAround 1665, Isaac Newton generalised the formula to allow real exponents other than nonnegative integers, and infact it can be generalised further, to complex exponents. In this generalisation, the finite sum is replaced by aninfinite series. In order to do this one needs to give meaning to binomial coefficients with an arbitrary upper index,which cannot be done using the above formula with factorials; however factoring out (n−k)! from numerator anddenominator in that formula, and replacing n by r which now stands for an arbitrary number, one can define

where is the Pochhammer symbol here standing for a falling factorial. Then, if x and y are real numbers with|x| > |y|,[7] and r is any complex number, one has

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Binomial theorem 21

When r is a nonnegative integer, the binomial coefficients for k > r are zero, so (2) specializes to (1), and there are atmost r + 1 nonzero terms. For other values of r, the series (2) has infinitely many nonzero terms, at least if x and yare nonzero.This is important when one is working with infinite series and would like to represent them in terms of generalisedhypergeometric functions.Taking r = −s leads to a useful but non-obvious formula:

Further specializing to s = 1 yields the geometric series formula.

Generalisations

Formula (2) can be generalised to the case where x and y are complex numbers. For this version, one should assume|x| > |y|[7] and define the powers of x + y and x using a holomorphic branch of log defined on an open disk of radius|x| centered at x.Formula (2) is valid also for elements x and y of a Banach algebra as long as xy = yx, x is invertible, and ||y/x|| < 1.

The multinomial theoremThe binomial theorem can be generalised to include powers of sums with more than two terms. The general versionis

where the summation is taken over all sequences of nonnegative integer indices k1 through km such that the sum ofall ki is n. (For each term in the expansion, the exponents must add up to n). The coefficients are known

as multinomial coefficients, and can be computed by the formula

Combinatorially, the multinomial coefficient counts the number of different ways to partition an

n-element set into disjoint subsets of sizes k1, ..., kn.

The multi-binomial theoremIt is often useful, when working in more dimension, to deal with products of binomial expressions. By the binomialtheorem this is equal toThis may be written more concisely, by multi-index notation, as

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Binomial theorem 22

Applications

Multiple angle identitiesFor the complex numbers the binomial theorem can be combined with De Moivre's formula to yield multiple-angleformulas for the sine and cosine. According to De Moivre's formula,

Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts canbe taken to yield formulas for cos(nx) and sin(nx). For example, since

De Moivre's formula tells us that

which are the usual double-angle identities. Similarly, since

De Moivre's formula yields

In general,

and

Series for eThe number e is often defined by the formula

Applying the binomial theorem to this expression yields the usual infinite series for e. In particular:

The kth term of this sum is

As n → ∞, the rational expression on the right approaches one, and therefore

This indicates that e can be written as a series:

Indeed, since each term of the binomial expansion is an increasing function of n, it follows from the monotoneconvergence theorem for series that the sum of this infinite series is equal to e.

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Binomial theorem 23

The binomial theorem in abstract algebraFormula (1) is valid more generally for any elements x and y of a semiring satisfying xy = yx. The theorem is trueeven more generally: alternativity suffices in place of associativity.The binomial theorem can be stated by saying that the polynomial sequence { 1, x, x2, x3, ... } is of binomial type.

Notes[1] Binomial Theorem (http:/ / mathworld. wolfram. com/ BinomialTheorem. html)[2] The Story of the Binomial Theorem, by J. L. Coolidge (http:/ / www. jstor. org/ pss/ 2305028), The American Mathematical Monthly 56:3

(1949), pp. 147–157[3] O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji" (http:/ / www-history. mcs. st-andrews. ac.

uk/ Biographies/ Al-Karaji. html), MacTutor History of Mathematics archive, University of St Andrews, .[4] Sandler, Stanley (2011). An Introduction to Applied Statistical Thermodynamics. Hoboken NJ: John Wiley & Sons, Inc..

ISBN 978-0-470-91347-5.[5] Landau, James A (1999-05-08). "Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle" (http:/ / archives. math. utk. edu/

hypermail/ historia/ may99/ 0073. html) (mailing list email). Archives of Historia Matematica. . Retrieved 2007-04-13.[6] (Barth 2004)[7] This is to guarantee convergence. Depending on r, the series may also converge sometimes when |x| = |y|.

References• Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". Indian J. History Sci 1 (1): 68–74.• Barth, Nils R. (November 2004). "Computing Cavalieri's Quadrature Formula by a Symmetry of the n-Cube". The

American Mathematical Monthly (Mathematical Association of America) 111 (9): 811–813.doi:10.2307/4145193. ISSN 0002-9890. JSTOR 4145193, author's copy (http:/ / nbarth. net/ math/ papers/barth-01-cavalieri. pdf), further remarks and resources (http:/ / nbarth. net/ math/ papers/ )

• Graham, Ronald; Donald Knuth, Oren Patashnik (1994). "(5) Binomial Coefficients". Concrete Mathematics (2nded.). Addison Wesley. pp. 153–256. ISBN 0-201-55802-5. OCLC 17649857.

• Solomentsev, E.D. (2001), "Newton binomial" (http:/ / eom. springer. de/ n/ n066500. htm), in Hazewinkel,Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

External links• Binomial Theorem (http:/ / demonstrations. wolfram. com/ BinomialTheorem/ ) by Stephen Wolfram, and

"Binomial Theorem (Step-by-Step)" (http:/ / demonstrations. wolfram. com/ BinomialTheoremStepByStep/ ) byBruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007.

• Binomial Theorem Introduction (http:/ / www. liftminds. com/ lesson/ 23/ Bionomial_theorem_introduction)This article incorporates material from inductive proof of binomial theorem on PlanetMath, which is licensed underthe Creative Commons Attribution/Share-Alike License.

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Birch's theorem 24

Birch's theoremIn mathematics, Birch's theorem,[1] named for Bryan John Birch, is a statement about the representability of zero byodd degree forms.

Statement of Birch's theoremLet K be an algebraic number field, k, l and n be natural numbers, r1,...,rk be odd natural numbers, and f1,...,fk behomogeneous polynomials with coefficients in K of degrees r1,...,rk respectively in n variables, then there exists anumber ψ(r1,...,rk,l,K) such that

implies that there exists an l-dimensional vector subspace V of Kn such that

RemarksThe proof of the theorem is by induction over the maximal degree of the forms f1,...,fk. Essential to the proof is aspecial case, which can be proved by an application of the Hardy-Littlewood circle method, of the theorem whichstates that if n is sufficiently large and r is odd, then the equation

has a solution in integers x1,...,xn, not all of which are 0.The restriction to odd r is necessary, since even-degree forms, such as positive definite quadratic forms, may take thevalue 0 only at the origin.

References[1] B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika, 4, pages 102-105 (1957)

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Birkhoff's representation theorem 25

Birkhoff's representation theoremThis is about lattice theory. For other similarly named results, see Birkhoff's theorem.

In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finitedistributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions andintersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributivelattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topologicalspaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.[1]

The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935on the representation of Boolean algebras as families of sets closed under union, intersection, and complement(so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), andBirkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theoremhas also been called the fundamental theorem for finite distributive lattices.[2]

Understanding the theoremMany lattices can be defined in such a way that the elements of the lattice are represented by sets, the join operationof the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. Forinstance, the Boolean lattice defined from the family of all subsets of a finite set has this property. More generallyany finite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obeythe distributive law, any lattice defined in this way is a distributive lattice. Birkhoff's theorem states that in fact allfinite distributive lattices can be obtained this way, and later generalizations of Birkhoff's theorem state the samething for infinite lattices.

Examples

The distributive lattice of divisors of 120, and its representation as sets of prime powers.

Consider the divisors of somecomposite number, such as (in thefigure) 120, partially ordered bydivisibility. Any two divisors of 120,such as 12 and 20, have a uniquegreatest common factor 12 ∧ 20 = 4,the largest number that divides both ofthem, and a unique least commonmultiple 12 ∨ 20 = 60; both of thesenumbers are also divisors of 120.These two operations ∨ and ∧ satisfythe distributive law, in either of two equivalent forms: (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z) and(x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z), for all x, y, and z. Therefore, the divisors form a finite distributive lattice.

One may associate each divisor with the set of prime powers that divide it: thus, 12 is associated with the set {2,3,4},while 20 is associated with the set {2,4,5}. Then 12 ∧ 20 = 4 is associated with the set {2,3,4} ∩ {2,4,5} = {2,4},while 12 ∨ 20 = 60 is associated with the set {2,3,4} ∪ {2,4,5} = {2,3,4,5}, so the join and meet operations of thelattice correspond to union and intersection of sets.The prime powers 2, 3, 4, 5, and 8 appearing as elements in these sets may themselves be partially ordered by divisibility; in this smaller partial order, 2 ≤ 4 ≤ 8 and there are no order relations between other pairs. The 16 sets that are associated with divisors of 120 are the lower sets of this smaller partial order, subsets of elements such that if

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Birkhoff's representation theorem 26

x ≤ y and y belongs to the subset, then x must also belong to the subset. From any lower set L, one can recover theassociated divisor by computing the least common multiple of the prime powers in L. Thus, the partial order on thefive prime powers 2, 3, 4, 5, and 8 carries enough information to recover the entire original 16-element divisibilitylattice.Birkhoff's theorem states that this relation between the operations ∧ and ∨ of the lattice of divisors and theoperations ∩ and ∪ of the associated sets of prime powers is not coincidental, and not dependent on the specificproperties of prime numbers and divisibility: the elements of any finite distributive lattice may be associated withlower sets of a partial order in the same way.As another example, the application of Birkhoff's theorem to the family of subsets of an n-element set, partiallyordered by inclusion, produces the free distributive lattice with n generators. The number of elements in this lattice isgiven by the Dedekind numbers.

The partial order of join-irreduciblesIn a lattice, an element x is join-irreducible if x is not the join of a finite set of other elements. Equivalently, x isjoin-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any twosmaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 isjoin-irreducible. An element x is join-prime if, whenever x ≤ y ∨ z, either x ≤ y or x ≤ z. In the same lattice, 4 isjoin-prime: whenever lcm(y,z) is divisible by 4, at least one of y and z must itself be divisible by 4.In any lattice, a join-prime element must be join-irreducible. Equivalently, an element that is not join-irreducible isnot join-prime. For, if an element x is not join-irreducible, there exist smaller y and z such that x = y ∨ z. But thenx ≤ y ∨ z, and x is not less than or equal to either y or z, showing that it is not join-prime.There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but in adistributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x ≤ y ∨ z. Thisinequality is equivalent to the statement that x = x ∧ (y ∨ z), and by the distributive law x = (x ∧ y) ∨ (x ∧ z). Butsince x is join-irreducible, at least one of the two terms in this join must be x itself, showing that either x = x ∧ y(equivalently x ≤ y) or x = x ∧ z (equivalently x ≤ z).The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhoff's theorem states that thelattice itself can be recovered from the lower sets of this partial order.

Birkhoff's theoremIn any partial order, the lower sets form a lattice in which the lattice's partial ordering is given by set inclusion, thejoin operation corresponds to set union, and the meet operation corresponds to set intersection, because unions andintersections preserve the property of being a lower set. Because set unions and intersections obey the distributivelaw, this is a distributive lattice. Birkhoff's theorem states that any finite distributive lattice can be constructed in thisway.

Theorem. Any finite distributive lattice L is isomorphic to the lattice of lower sets of the partial order of thejoin-irreducible elements of L.

That is, there is a one-to-one order-preserving correspondence between elements of L and lower sets of the partialorder. The lower set corresponding to an element x of L is simply the set of join-irreducible elements of L that areless than or equal to x, and the element of L corresponding to a lower set S of join-irreducible elements is the join ofS.If one starts with a lower set S, lets x be the join of S, and constructs lower set T of the join-irreducible elements less than or equal to x, then S = T. For, every element of S clearly belongs to T, and any join-irreducible element less than or equal to x must (by join-primality) be less than or equal to one of the members of S, and therefore must (by the assumption that S is a lower set) belong to S itself. Conversely, if one starts with an element x of L, lets S be the

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join-irreducible elements less than or equal to x, and constructs y as the join of S, then x = y. For, as a join ofelements less than or equal to x, y can be no greater than x itself, but if x is join-irreducible then x belongs to S whileif x is the join of two or more join-irreducible items then they must again belong to S, so y ≥ x. Therefore, thecorrespondence is one-to-one and the theorem is proved.

Rings of sets and preordersBirkhoff (1937) defined a ring of sets to be a family of sets that is closed under the operations of set unions and setintersections; later, motivated by applications in mathematical psychology, Doignon & Falmagne (1999) called thesame structure a quasi-ordinal knowledge space. If the sets in a ring of sets are ordered by inclusion, they form adistributive lattice. The elements of the sets may be given a preorder in which x ≤ y whenever some set in the ringcontains x but not y. The ring of sets itself is then the family of lower sets of this preorder, and any preorder givesrise to a ring of sets in this way.

FunctorialityBirkhoff's theorem, as stated above, is a correspondence between individual partial orders and distributive lattices.However, it can also be extended to a correspondence between order-preserving functions of partial orders andbounded homomorphisms of the corresponding distributive lattices. The direction of these maps is reversed in thiscorrespondence.Let 2 denote the partial order on the two-element set {0, 1}, with the order relation 0 < 1, and (following Stanley) letJ(P) denote the distributive lattice of lower sets of a finite partial order P. Then the elements of J(P) correspondone-for-one to the order-preserving functions from P to 2.[2] For, if ƒ is such a function, ƒ−1(0) forms a lower set, andconversely if L is a lower set one may define an order-preserving function ƒL that maps L to 0 and that maps theremaining elements of P to 1. If g is any order-preserving function from Q to P, one may define a function g* fromJ(P) to J(Q) that uses the composition of functions to map any element L of J(P) to ƒL ∘ g. This composite functionmaps Q to 2 and therefore corresponds to an element g*(L) = (ƒL ∘ g)−1(0) of J(Q). Further, for any x and y in J(P),g*(x ∧ y) = g*(x) ∧ g*(y) (an element of Q is mapped by g to the lower set x ∩ y if and only if belongs both to the setof elements mapped to x and the set of elements mapped to y) and symmetrically g*(x ∨ y) = g*(x) ∨ g*(y).Additionally, the bottom element of J(P) (the function that maps all elements of P to 0) is mapped by g* to thebottom element of J(Q), and the top element of J(P) is mapped by g* to the top element of J(Q). That is, g* is ahomomorphism of bounded lattices.However, the elements of P themselves correspond one-for-one with bounded lattice homomorphisms from J(P) to2. For, if x is any element of P, one may define a bounded lattice homomorphism jx that maps all lower setscontaining x to 1 and all other lower sets to 0. And, for any lattice homomorphism from J(P) to 2, the elements ofJ(P) that are mapped to 1 must have a unique minimal element x (the meet of all elements mapped to 1), which mustbe join-irreducible (it cannot be the join of any set of elements mapped to 0), so every lattice homomorphism has theform jx for some x. Again, from any bounded lattice homomorphism h from J(P) to J(Q) one may use composition offunctions to define an order-preserving map h* from Q to P. It may be verified that g** = g for any order-preservingmap g from Q to P and that and h** = h for any bounded lattice homomorphism h from J(P) to J(Q).In category theoretic terminology, J is a contravariant hom-functor J = Hom(—,2) that defines a duality of categoriesbetween, on the one hand, the category of finite partial orders and order-preserving maps, and on the other hand thecategory of finite distributive lattices and bounded lattice homomorphisms.

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GeneralizationsIn an infinite distributive lattice, it may not be the case that the lower sets of the join-irreducible elements are inone-to-one correspondence with lattice elements. Indeed, there may be no join-irreducibles at all. This happens, forinstance, in the lattice of all integers, ordered with the reverse of the usual divisibility ordering (so x ≤ y when ydivides x): any number x can be expressed as the join of numbers xp and xq where p and q are distinct prime numbersthat do not divide x. However, elements in infinite distributive lattices may still be represented as sets via Stone'srepresentation theorem for distributive lattices, a form of Stone duality in which each lattice element corresponds toa compact open set in a certain topological space. This generalized representation theorem can be expressed as acategory-theoretic duality between distributive lattices and coherent spaces (sometimes called spectral spaces),topological spaces in which the compact open sets are closed under intersection and form a base for the topology.[3]

Hilary Priestley showed that Stone's representation theorem could be interpreted as an extension of the idea ofrepresenting lattice elements by lower sets of a partial order, using Nachbin's idea of ordered topological spaces.Stone spaces with an additional partial order linked with the topology via Priestley separation axiom can also be usedto represent bounded distributive lattices. Such spaces are known as Priestley spaces. Further, certain bitopologicalspaces, namely pairwise Stone spaces, generalize Stone's original approach by utilizing two topologies on a set torepresent an abstract distributve lattice. Thus, Birkhoff's representation theorem extends to the case of infinite(bounded) distributive lattices in at least three different ways, summed up in duality theory for distributive lattices.Birkhoff's representation theorem may also be generalized to finite structures other than distributive lattices. In adistributive lattice, the self-dual median operation[4]

gives rise to a median algebra, and the covering relation of the lattice forms a median graph. Finite median algebrasand median graphs have a dual structure as the set of solutions of a 2-satisfiability instance; Barthélemy &Constantin (1993) formulate this structure equivalently as the family of initial stable sets in a mixed graph.[5] For adistributive lattice, the corresponding mixed graph has no undirected edges, and the initial stable sets are just thelower sets of the transitive closure of the graph. Equivalently, for a distributive lattice, the implication graph of the2-satisfiability instance can be partitioned into two connected components, one on the positive variables of theinstance and the other on the negative variables; the transitive closure of the positive component is the underlyingpartial order of the distributive lattice.Another result analogous to Birkhoff's representation theorem, but applying to a broader class of lattices, is thetheorem of Edelman (1980) that any finite join-distributive lattice may be represented as an antimatroid, a family ofsets closed under unions but in which closure under intersections has been replaced by the property that eachnonempty set has a removable element.

Notes[1] Birkhoff (1937).[2] (Stanley 1997).[3] Johnstone (1982).[4] Birkhoff & Kiss (1947).[5] A minor difference between the 2-SAT and initial stable set formulations is that the latter presupposes the choice of a fixed base point from

the median graph that corresponds to the empty initial stable set.

References• Barthélemy, J.-P.; Constantin, J. (1993), "Median graphs, parallelism and posets", Discrete Mathematics 111

(1–3): 49–63, doi:10.1016/0012-365X(93)90140-O.• Birkhoff, Garrett (1937), "Rings of sets", Duke Mathematical Journal 3 (3): 443–454,

doi:10.1215/S0012-7094-37-00334-X.

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• Birkhoff, Garrett; Kiss, S. A. (1947), "A ternary operation in distributive lattices" (http:/ / projecteuclid. org/euclid. bams/ 1183510977), Bulletin of the American Mathematical Society 52 (1): 749–752, MR0021540.

• Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.• Edelman, Paul H. (1980), "Meet-distributive lattices and the anti-exchange closure", Algebra Universalis 10 (1):

290–299, doi:10.1007/BF02482912.• Johnstone, Peter (1982), "II.3 Coherent locales", Stone Spaces, Cambridge University Press, pp. 62–69,

ISBN 9780521337793.• Priestley, H. A. (1970), "Representation of distributive lattices by means of ordered Stone spaces", Bulletin of the

London Mathematical Society 2 (2): 186–190, doi:10.1112/blms/2.2.186.• Priestley, H. A. (1972), "Ordered topological spaces and the representation of distributive lattices", Proceedings

of the London Mathematical Society 24 (3): 507–530, doi:10.1112/plms/s3-24.3.507.• Stanley, R. P. (1997), Enumerative Combinatorics, Volume I, Cambridge Studies in Advanced Mathematics 49,

Cambridge University Press, pp. 104–112.

Boolean prime ideal theoremIn mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstractalgebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra canbe extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Othertheorems are obtained by considering different mathematical structures with appropriate notions of ideals, forexample, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). Thisarticle focuses on prime ideal theorems from order theory.Although the various prime ideal theorems may appear simple and intuitive, they cannot be derived in general fromthe axioms of Zermelo–Fraenkel set theory (ZF). Instead, some of the statements turn out to be equivalent to theaxiom of choice (AC), while others—the Boolean prime ideal theorem, for instance—represent a property that isstrictly weaker than AC. It is due to this intermediate status between ZF and ZF + AC (ZFC) that the Boolean primeideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) aresometimes used to refer to this additional axiom.

Prime ideal theoremsRecall that an order ideal is a (non-empty) directed lower set. If the considered poset has binary suprema (a.k.a.joins), as do the posets within this article, then this is equivalently characterized as a lower set I which is closed forbinary suprema (i.e. x, y in I imply x y in I). An ideal I is prime if, whenever an infimum x y is in I, one alsohas x in I or y in I. Ideals are proper if they are not equal to the whole poset.Historically, the first statement relating to later prime ideal theorems was in fact referring to filters—subsets that areideals with respect to the dual order. The ultrafilter lemma states that every filter on a set is contained within somemaximal (proper) filter—an ultrafilter. Recall that filters on sets are proper filters of the Boolean algebra of itspowerset. In this special case, maximal filters (i.e. filters that are not strict subsets of any proper filter) and primefilters (i.e. filters that with each union of subsets X and Y contain also X or Y) coincide. The dual of this statementthus assures that every ideal of a powerset is contained in a prime ideal.The above statement led to various generalized prime ideal theorems, each of which exists in a weak and in a strong form. Weak prime ideal theorems state that every non-trivial algebra of a certain class has at least one prime ideal. In contrast, strong prime ideal theorems require that every ideal that is disjoint from a given filter can be extended to a prime ideal which is still disjoint from that filter. In the case of algebras that are not posets, one uses different substructures instead of filters. Many forms of these theorems are actually known to be equivalent, so that the

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assertion that "PIT" holds is usually taken as the assertion that the corresponding statement for Boolean algebras(BPI) is valid.Another variation of similar theorems is obtained by replacing each occurrence of prime ideal by maximal ideal. Thecorresponding maximal ideal theorems (MIT) are often—though not always—stronger than their PIT equivalents.

Boolean prime ideal theoremThe Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statementis:

Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I iscontained in some prime ideal of B that is disjoint from F.

The weak prime ideal theorem for Boolean algebras simply states:Every Boolean algebra contains a prime ideal.

We refer to these statements as the weak and strong BPI. The two are equivalent, as the strong BPI clearly impliesthe weak BPI, and the reverse implication can be achieved by using the weak BPI to find prime ideals in theappropriate quotient algebra.The BPI can be expressed in various different ways. For this purpose, recall the following theorem:For any ideal I of a Boolean algebra B, the following are equivalent:• I is a prime ideal.• I is a maximal proper ideal, i.e. for any proper ideal J, if I is contained in J then I = J.• For every element a of B, I contains exactly one of {a, ¬a}.This theorem is a well-known fact for Boolean algebras. Its dual establishes the equivalence of prime filters andultrafilters. Note that the last property is in fact self-dual—only the prior assumption that I is an ideal gives the fullcharacterization. It is worth mentioning that all of the implications within this theorem can be proven in classicalZermelo-Fraenkel set theory.Thus the following (strong) maximal ideal theorem (MIT) for Boolean algebras is equivalent to BPI:

Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I iscontained in some maximal ideal of B that is disjoint from F.

Note that one requires "global" maximality, not just maximality with respect to being disjoint from F. Yet, thisvariation yields another equivalent characterization of BPI:

Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I iscontained in some ideal of B that is maximal among all ideals disjoint from F.

The fact that this statement is equivalent to BPI is easily established by noting the following theorem: For anydistributive lattice L, if an ideal I is maximal among all ideals of L that are disjoint to a given filter F, then I is aprime ideal. The proof for this statement (which can again be carried out in ZF set theory) is included in the articleon ideals. Since any Boolean algebra is a distributive lattice, this shows the desired implication.All of the above statements are now easily seen to be equivalent. Going even further, one can exploit the fact thedual orders of Boolean algebras are exactly the Boolean algebras themselves. Hence, when taking the equivalentduals of all former statements, one ends up with a number of theorems that equally apply to Boolean algebras, butwhere every occurrence of ideal is replaced by filter. It is worth noting that for the special case where the Booleanalgebra under consideration is a powerset with the subset ordering, the "maximal filter theorem" is called theultrafilter lemma.Summing up, for Boolean algebras, the weak and strong MIT, the weak and strong PIT, and these statements with filters in place of ideals are all equivalent. It is known that all of these statements are consequences of the axiom of choice (the easy proof makes use of Zorn's lemma), but cannot be proven in classical Zermelo-Fraenkel set theory.

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Yet, the BPI is strictly weaker than the axiom of choice, though the proof of this statement, due to J. D. Halpern andAzriel Levy is rather non-trivial.

Further prime ideal theoremsThe prototypical properties that were discussed for Boolean algebras in the above section can easily be modified toinclude more general lattices, such as distributive lattices or Heyting algebras. However, in these cases maximalideals are different from prime ideals, and the relation between PITs and MITs is not obvious.Indeed, it turns out that the MITs for distributive lattices and even for Heyting algebras are equivalent to the axiomof choice. On the other hand, it is known that the strong PIT for distributive lattices is equivalent to BPI (i.e. to theMIT and PIT for Boolean algebras). Hence this statement is strictly weaker than the axiom of choice. Furthermore,observe that Heyting algebras are not self dual, and thus using filters in place of ideals yields different theorems inthis setting. Maybe surprisingly, the MIT for the duals of Heyting algebras is not stronger than BPI, which is in sharpcontrast to the abovementioned MIT for Heyting algebras.Finally, prime ideal theorems do also exist for other (not order-theoretical) abstract algebras. For example, the MITfor rings implies the axiom of choice. This situation requires to replace the order-theoretic term "filter" by otherconcepts—for rings a "multiplicatively closed subset" is appropriate.

The ultrafilter lemmaA filter on a set X is a collection of nonempty subsets of X that is closed under finite intersection and under superset.An ultrafilter is a maximal filter. The ultrafilter lemma states that every filter on a set X is a subset of someultrafilter on X (a maximal filter of nonempty subsets of X).[1] This lemma is most often used in the study oftopology. The existence of non-principal ultrafilters is due to Tarski in 1930.The ultrafilter lemma is equivalent to the Boolean prime ideal theorem, with the equivalence provable in ZF settheory without the axiom of choice. The idea behind the proof is that the subsets of any set form a Boolean algebrapartially ordered by inclusion, and any Boolean algebra is representable as an algebra of sets by Stone'srepresentation theorem.

ApplicationsIntuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in thesense that we can extend every ideal to a maximal one. This is of practical importance for proving Stone'srepresentation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all primeideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data.Furthermore, it turns out that in applications one can freely choose either to work with prime ideals or with primefilters, because every ideal uniquely determines a filter: the set of all Boolean complements of its elements. Bothapproaches are found in the literature.Many other theorems of general topology that are often said to rely on the axiom of choice are in fact equivalent toBPI. For example, the theorem that a product of compact Hausdorff spaces is compact is equivalent to it. If we leaveout "Hausdorff" we get a theorem equivalent to the full axiom of choice.A not too well known application of the Boolean prime ideal theorem is the existence of a non-measurable set[2] (theexample usually given is the Vitali set, which requires the Axiom of Choice). From this and the fact that the BPI isstrictly weaker than the Axiom of Choice, it follows that the existence of non-measurable sets is strictly weaker thanthe axiom of choice.

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Notes[1] Halpern, James D. (1966), "Bases in Vector Spaces and the Axiom of Choice", Proceedings of the American Mathematical Society (American

Mathematical Society) 17 (3): 670–673, JSTOR 2035388.[2] Sierpiński, Wacław (1938), "Fonctions additives non complètement additives et fonctions non mesurables", Fundamenta Mathematicae 30:

96–99

References• Davey, B. A.; Priestley, H. A. (2002), Introduction to Lattices and Order (2nd ed.), Cambridge University Press,

ISBN 9780521784511.An easy to read introduction, showing the equivalence of PIT for Boolean algebras and distributive lattices.

• Johnstone, Peter (1982), Stone Spaces, Cambridge studies in advanced mathematics, 3, Cambridge UniversityPress, ISBN 9780521337793.

The theory in this book often requires choice principles. The notes on various chapters discuss the generalrelation of the theorems to PIT and MIT for various structures (though mostly lattices) and give pointers tofurther literature.

• Banaschewski, B. (1983), "The power of the ultrafilter theorem", Journal of the London Mathematical Society(2nd series) 27 (2): 193–202, doi:10.1112/jlms/s2-27.2.193.

Discusses the status of the ultrafilter lemma.

• Erné, M. (2000), "Prime ideal theory for general algebras", Applied Categorical Structures 8: 115–144.Gives many equivalent statements for the BPI, including prime ideal theorems for other algebraic structures.PITs are considered as special instances of separation lemmas.

Borel–Weil theoremIn mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel andAndré Weil, provides a concrete model for irreducible representations of compact Lie groups and complexsemisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic linebundles on the flag manifold of the group. Its generalization to higher cohomology spaces is called theBorel–Weil–Bott theorem.

Statement of the theoremThe theorem can be stated either for a complex semisimple Lie group G or for its compact form K. Let G be aconnected complex semisimple Lie group, B its Borel subgroup, and X=G/B the flag variety. In this picture, X is acomplex manifold and a nonsingular algebraic G-variety. The flag variety can also be described as a compacthomogeneous space K/T, where T=K∩B is a (compact) Cartan subgroup of K. An integral weight λ determines aG-equivariant holomorphic line bundle Lλ on X and the group G acts on its space of global sections,

The Borel–Weil theorem states that if λ is a dominant integral weight then this representation is an irreduciblehighest weight representation of G with highest weight λ. Its restriction to K is an irreducible unitary representationof K with highest weight λ, and each irreducible unitary representations of K is obtained in this way for a uniquevalue of λ.

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Concrete descriptionThe weight λ gives rise to a character (one-dimensional representation) of the Borel subgroup B, which is denoted χλ.Holomorphic sections of the holomorphic line bundle Lλ over G/B may be described more concretely as holomorphicmaps

for all g∈G and b∈B.The action of G on these sections is given by

for g,h∈G.

ExampleLet G be the complex special linear group SL(2,C), with a Borel subgroup consisting of upper triangular matriceswith determinant one. Integral weights for G may be identified with integers, with dominant weights correspondingto nonnegative integers, and the corresponding characters χn of B have the form

The flag variety G/B may be identified with the complex projective line P1 with homogeneous coordinates X, Y andthe space of the global sections of the line bundle Ln is identified with the space of homogeneous polynomials ofdegree n on C2. For n≥0, this space has dimension n+1 and forms an irreducible representation under the standardaction of G on the polynomial algebra C[X,Y]. Weight vectors are given by monomials

of weights 2i−n, and the highest weight vector Xn has weight n.

HistoryThe theorem dates back to the early 1950s and can be found in Serre (1995) and Tits (1955).

References• Serre, Jean-Pierre (1995), "Représentations linéaires et espaces homogènes kählériens des groupes de Lie

compacts (d'après Armand Borel et André Weil)", Séminaire Bourbaki (Paris: Soc. Math. France) 2 (100):447–454. In French; translated title: “Linear representations and Kähler homogeneous spaces of compact Liegroups (after Armand Borel and André Weil.”

• Tits, Jacques (1955), Sur certaines classes d'espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci.Mém. Coll., 29 In French.

• Sepanski, Mark R. (2007), Compact Lie groups., Graduate Texts in Mathematics, 235, New York: Springer.• Knapp, Anthony W. (2001), Representation theory of semisimple groups: An overview based on examples,

Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press. Reprint of the 1986 original.

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Borel–Weil–Bott theoremIn mathematics, the Borel–Weil-Bott theorem is a basic result in the representation theory of Lie groups, showinghow a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and,more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlierBorel–Weil theorem of Armand Borel and André Weil, dealing just with the section case, the extension beingprovided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraicgeometry in the Zariski topology.

FormulationLet G be a semisimple Lie group or algebraic group over , and fix a maximal torus T along with a Borelsubgroup B which contains T. Let λ be an integral weight of T; λ defines in a natural way a one-dimensionalrepresentation Cλ of B, by pulling back the representation on T = B/U, where U is the unipotent radical of B. Sincewe can think of the projection map G → G/B as a principal B-bundle, for each Cλ we get an associated fiber bundleL-λ on G/B (note the sign), which is obviously a line bundle. Identifying Lλ with its sheaf of holomorphic sections,we consider the sheaf cohomology groups . Since G acts on the total space of the bundle bybundle automorphisms, this action naturally gives a G-module structure on these groups; and the Borel–Weil–Botttheorem gives an explicit description of these groups as G-modules.We first need to describe the Weyl group action centered at . For any integral weight and in the Weyl groupW, we set , where denotes the half-sum of positive roots of G. It is straightforward tocheck that this defines a group action, although this action is not linear, unlike the usual Weyl group action. Also, aweight is said to be dominant if for all simple roots . Let denote the length function on W.Given an integral weight , one of two cases occur: (1) There is no such that is dominant,equivalently, there exists a nonidentity such that ; or (2) There is a unique suchthat is dominant. The theorem states that in the first case, we have

for all i;and in the second case, we have

for all , whileis the dual of the irreducible highest-weight representation of G with highest weight .

It is worth noting that case (1) above occurs if and only if for some positive root . Also, we obtainthe classical Borel–Weil theorem as a special case of this theorem by taking to be dominant and to be theidentity element .

ExampleFor example, consider G = SL2(C), for which G/B is the Riemann sphere, an integral weight is specified simply byan integer n, and ρ = 1. The line bundle Ln is O(n), whose sections are the homogeneous polynomials of degree n(i.e. the binary forms). As a representation of G, the sections can be written as Symn(C2)*, and is canonicallyisomorphic to Symn(C2). This gives us at a stroke the representation theory of : Γ(O(1)) is the standardrepresentation, and Γ(O(n)) is its n-th symmetric power. We even have a unified description of the action of the Liealgebra, derived from its realization as vector fields on the Riemann sphere: if H, X, Y are the standard generators of

, then we can write

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Positive characteristicOne also has a weaker form of this theorem in positive characteristic. Namely, let G be a semisimple algebraic groupover an algebraically closed field of characteristic . Then it remains true that for all iif is a weight such that is non-dominant for all . However, the other statements of the theoremdo not remain valid in this setting.More explicitly, let be a dominant integral weight; then it is still true that for all ,but it is no longer true that this G-module is simple in general, although it does contain the unique highest weightmodule of highest weight as a G-submodule. If is an arbitrary integral weight, it is in fact a large unsolvedproblem in representation theory to describe the cohomology modules in general. Unlike over ,it need not be the case for a fixed that these modules are all zero except in a single degree i.

References• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics,

Readings in Mathematics, 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR1153249, ISBN978-0-387-97527-6.

• Baston, Robert J.; Eastwood, Michael G. (1989), The Penrose Transform: its Interaction with RepresentationTheory, Oxford University Press.

• Hazewinkel, Michiel, ed. (2001), "Bott–Borel–Weil theorem" [1], Encyclopaedia of Mathematics, Springer,ISBN 978-1556080104

• A Proof of the Borel–Weil–Bott Theorem [2], by Jacob Lurie. Retrieved on Dec. 14, 2007.This article incorporates material from Borel–Bott–Weil theorem on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

References[1] http:/ / eom. springer. de/ b/ b120400. htm[2] http:/ / www-math. mit. edu/ ~lurie/ papers/ bwb. pdf

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Brauer's theorem on induced charactersBrauer's theorem on induced characters, often known as Brauer's induction theorem, and named after RichardBrauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of therepresentation theory of a finite group. Let G be a finite group and let Char(G) denote the subring of the ring ofcomplex-valued class functions of G consisting of integer combinations of irreducible characters. Char(G) is knownas the character ring of G, and its elements are known as virtual characters (alternatively, as generalizedcharacters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of Gis again a character of G. Its multiplication is given by the elementwise product of class functions.Brauer's induction theorem shows that the character ring can be generated (as an abelian group) by certain characterswhich are fairly easily understood. More precisely, the theorem states that every virtual character of G is expressibleas an integer combination of induced characters of the form , , where H ranges over subgroups of G and λranges over linear characters (having degree 1) of H.In fact, Brauer showed that the subgroups H could be chosen from a very restricted collection, now called Brauerelementary subgroups. These are direct products of cyclic groups and groups whose order is a power of a prime.Using Frobenius reciprocity, Brauer's induction theorem leads easily to his fundamental characterization ofcharacters, which asserts that a complex-valued class function of G is a virtual character if and only if its restrictionto each Brauer elementary subgroup of G is a virtual character. This result, together with the fact that a virtualcharacter θ is an irreducible character if and only if θ(1) > 0 and (where is the usual inner producton the ring of complex-valued class functions) gives a means of constructing irreducible characters without explicitlyconstructing the associated representations.An initial motivation for Brauer's induction theorem was application to Artin L-functions. It shows that those arebuilt up from Dirichlet L-functions, or more general Hecke L-functions. Highly significant for that application iswhether each character of G is a non-negative integer combination of characters induced from linear characters ofsubgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of G are so expressible,then G must be a solvable group (although solvability alone does not guarantee such expressions- for example, thesolvable group SL(2,3) has an irreducible complex character of degree 2 which is not expressible as a non-negativeinteger combination of characters induced from linear characters of subgroups). An ingredient of the proof ofBrauer's induction theorem is that when G is a finite nilpotent group, every complex irreducible character of G isinduced from a linear character of some subgroup.A precursor to Brauer's induction theorem was Artin's induction theorem, which states that |G| times the trivialcharacter of G is an integer combination of characters which are each induced from trivial characters of cyclicsubgroups of G. Brauer's theorem removes the factor |G|, but at the expense of expanding the collection of subgroupsused. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such inductiontheorem (with integer combinations of characters induced from linear characters) could be proved with a collectionof subgroups smaller than the Brauer elementary subgroups.The proof of Brauer's induction theorem exploits the ring structure of Char(G) (most proofs also make use of a slightly larger ring, Char*(G), which consists of -combinations of irreducible characters, where ω is a primitive complex |G|-th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal I(G) of Char(G), so the proof reduces to showing that the trivial character is in I(G). Several proofs of the theorem, beginning with a proof due to Brauer and John Tate, show that the trivial character is in the analogously defined ideal I*(G) of Char*(G) by concentrating attention on one prime p at a time, and constructing integer-valued elements of I*(G) which differ (elementwise) from the trivial character by (integer multiples of) a sufficiently high power of p. Once this is achieved for every prime divisor of |G|, some manipulations with congruences and algebraic integers, again exploiting the fact that I*(G) is an ideal of Ch*(G), place the trivial

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character in I(G). An auxiliary result here is that a -valued class function lies in the ideal I*(G) if its values are alldivisible (in ) by |G|.Brauer's induction theorem was proved in 1946, and there are now many alternative proofs. In 1986, Victor Snaithgave a proof by a radically different approach, topological in nature (an application of the Lefschetz fixed-pointtheorem). There has been related recent work on the question of finding natural and explicit forms of Brauer'stheorem, notably by Robert Boltje.

References• Isaacs, I.M. (1994). Character Theory of Finite Groups. Dover. ISBN 0-486-68014-2. Corrected reprint of the

1976 original, published by Academic Press.

Brauer's three main theoremsBrauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finitegroup (in characteristic p) with those of its p-local subgroups, that is to say, the normalizers of its non-trivialp-subgroups.The second and third main theorems allow refinements of orthogonality relations for ordinary characters which maybe applied in finite group theory. These do not presently admit a proof purely in terms of ordinary characters. Allthree main theorems are stated in terms of the Brauer correspondence.

Brauer correspondenceThere are many ways to extend the definition which follows, but this is close to the early treatments by Brauer. Let Gbe a finite group, p be a prime, F be a field of characteristic p. Let H be a subgroup of G which contains

for some p-subgroup Q of G, and is contained in the normalizer

.The Brauer homomorphism (with respect to H) is a linear map from the center of the group algebra of G over F tothe corresponding algebra for H. Specifically, it is the restriction to of the (linear) projection from to

whose kernel is spanned by the elements of G outside . The image of this map is contained in, and it transpires that the map is also a ring homomorphism.

Since it is a ring homomorphism, for any block B of FG, the Brauer homomorphism sends the identity element of Beither to 0 or to an idempotent element. In the latter case, the idempotent may be decomposed as a sum of (mutuallyorthogonal) primitive idempotents of Z(FH). Each of these primitive idempotents is the multiplicative identity ofsome block of FH. The block b of FH is said to be a Brauer correspondent of B if its identity element occurs in thisdecomposition of the image of the identity of B under the Brauer homomorphism.

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Brauer's first main theoremBrauer's first main theorem (Brauer 1944, 1956, 1970) states that if is a finite group a is a -subgroup of

, then there is a bijection between the collections of (characteristic p) blocks of with defect group andblocks of the normalizer with defect group D. This bijection arises because when , eachblock of G with defect group D has a unique Brauer correspondent block of H, which also has defect group D.

Brauer's second main theoremBrauer's second main theorem (Brauer 1944, 1959) gives, for an element t whose order is a power of a prime p, acriterion for a (characteristic p) block of to correspond to a given block of , via generalized

decomposition numbers. These are the coefficients which occur when the restrictions of ordinary characters of (from the given block) to elements of the form tu, where u ranges over elements of order prime to p in , arewritten as linear combinations of the irreducible Brauer characters of . The content of the theorem is that it isonly necessary to use Brauer characters from blocks of which are Brauer correspondents of the chosen blockof G.

Brauer's third main theoremBrauer's third main theorem (Brauer 1964, theorem3) states that when Q is a p-subgroup of the finite group G, and His a subgroup of G, containing , and contained in , then the principal block of H is the onlyBrauer correspondent of the principal block of G (where the blocks referred to are calculated in characteristic p).

References• Brauer, R. (1944), "On the arithmetic in a group ring", Proceedings of the National Academy of Sciences of the

United States of America 30: 109–114, ISSN 0027-8424, JSTOR 87919, MR0010547• Brauer, R. (1946), "On blocks of characters of groups of finite order I", Proceedings of the National Academy of

Sciences of the United States of America 32: 182–186, ISSN 0027-8424, JSTOR 87578, MR0016418• Brauer, R. (1946), "On blocks of characters of groups of finite order. II", Proceedings of the National Academy of

Sciences of the United States of America 32: 215–219, ISSN 0027-8424, JSTOR 87838, MR0017280• Brauer, R. (1956), "Zur Darstellungstheorie der Gruppen endlicher Ordnung", Mathematische Zeitschrift 63:

406–444, doi:10.1007/BF01187950, ISSN 0025-5874, MR0075953• Brauer, R. (1959), "Zur Darstellungstheorie der Gruppen endlicher Ordnung. II", Mathematische Zeitschrift 72:

25–46, doi:10.1007/BF01162934, ISSN 0025-5874, MR0108542• Brauer, R. (1964), "Some applications of the theory of blocks of characters of finite groups. I", Journal of Algebra

1: 152–167, doi:10.1016/0021-8693(64)90031-6, ISSN 0021-8693, MR0168662• Brauer, R. (1970), "On the first main theorem on blocks of characters of finite groups." [1], Illinois Journal of

Mathematics 14: 183–187, ISSN 0019-2082, MR0267010• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman,

Graham, Finite simple groups. Proceedings of an Instructional Conference organized by the LondonMathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: AcademicPress, pp. 249–327, ISBN 978-0-12-563850-0, MR0360785 gives a detailed proof of the Brauer's main theorems.

• Ellers, H. (2001), "Brauer's first main theorem" [2], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,Springer, ISBN 978-1556080104

• Ellers, H. (2001), "Brauer height-zero conjecture" [3], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,Springer, ISBN 978-1556080104

• Ellers, H. (2001), "Brauer's second main theorem" [4], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,Springer, ISBN 978-1556080104

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• Ellers, H. (2001), "Brauer's third main theorem" [5], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,Springer, ISBN 978-1556080104

• Walter Feit, The representation theory of finite groups. North-Holland Mathematical Library, 25. North-HollandPublishing Co., Amsterdam-New York, 1982. xiv+502 pp. ISBN 0-444-86155-6

References[1] http:/ / projecteuclid. org/ euclid. ijm/ 1256053174[2] http:/ / eom. springer. de/ b/ b120440. htm[3] http:/ / eom. springer. de/ b/ b120450. htm[4] http:/ / eom. springer. de/ b/ b120460. htm[5] http:/ / eom. springer. de/ b/ b120470. htm

Brauer–Cartan–Hua theoremThe Brauer–Cartan–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem inabstract algebra pertaining to division rings, which says that given two division rings K ≤ D such that xKx−1 iscontained in K for every x not equal to 0 in D, then either K is contained in Z, the center of D, or K = D. In otherwords, if the unit group of K is a normal subgroup of the unit group of D, then either K = D or K is central, (Lam2001, p. 211).

References• Herstein, I. N. (1975). Topics in algebra. New York: Wiley. p. 368. ISBN 0-471-01090-1.• Lam, Tsit-Yuen (2001). A First Course in Noncommutative Rings (2nd ed.). Berlin, New York: Springer-Verlag.

ISBN 978-0-387-95325-0. MR1838439.

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BrauerNesbitt theorem 40

Brauer–Nesbitt theoremIn mathematics, the Brauer-Nesbitt theorem can refer to several different theorems proved by Richard Brauer andCecil J. Nesbitt in the representation theory of finite groups.In modular representation theory, the Brauer-Nesbitt theorem on blocks of defect zero states that a characterwhose order is divisible by the highest power of a prime p dividing the order of a finite group remains irreduciblewhen reduced mod p and vanishes on all elements whose order is divisible by p. Moreover it belongs to a block ofdefect zero. A block of defect zero contains only one ordinary character and only one modular character.Another version states that if k is a field of characteristic zero, A is a k-algebra, V, W are semisimple A-moduleswhich are finite dimensional over k, and TrV = TrW as elements of Homk(A,k), then V and W are isomorphic asA-modules.

References• Curtis, Reiner, Representation theory of finite groups and associative algebras, Wiley 1962.• Brauer, R.; Nesbitt, C. On the modular characters of groups. Ann. of Math. (2) 42, (1941). 556-590.

Brauer–Siegel theoremIn mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptoticresult on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attemptsto generalise the results known on the class numbers of imaginary quadratic fields, to a more general sequence ofnumber fields

In all cases other than the rational field Q and imaginary quadratic fields, the regulator Ri of Ki must be taken intoaccount, because Ki then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of thestandard Brauer–Siegel theorem is that if Di is the discriminant of Ki, then

Assuming that, and the algebraic hypothesis that Ki is a Galois extension of Q, the conclusion is that

where hi is the class number of Ki.This result is ineffective, as indeed was the result on quadratic fields on which it built. Effective results in the samedirection were initiated in work of Harold Stark from the early 1970s.

References• Richard Brauer, On the Zeta-Function of Algebraic Number Fields, American Journal of Mathematics 69 (1947),

243–250.

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BrauerSuzuki theorem 41

Brauer–Suzuki theoremIn mathematics, the Brauer–Suzuki theorem, proved by Brauer & Suzuki (1959), Suzuki (1962), Brauer (1964),states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups ofodd order, then the group has a centre of order 2. In particular, such a group cannot be simple.A generalization of the Brauer–Suzuki theorem is given by Glauberman's Z* theorem.

References• Brauer, R. (1964), "Some applications of the theory of blocks of characters of finite groups. II", Journal of

Algebra 1: 307–334, doi:10.1016/0021-8693(64)90011-0, ISSN 0021-8693, MR0174636• Brauer, R.; Suzuki, Michio (1959), "On finite groups of even order whose 2-Sylow group is a quaternion group",

Proceedings of the National Academy of Sciences of the United States of America 45: 1757–1759,ISSN 0027-8424, JSTOR 90063, MR0109846

• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman,Graham, Finite simple groups. Proceedings of an Instructional Conference organized by the LondonMathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: AcademicPress, pp. 249–327, ISBN 978-0-12-563850-0, MR0360785 gives a detailed proof of the Brauer–Suzuki theorem.

• Suzuki, Michio (1962), "Applications of group characters" [1], in Hall, M., 1960 Institute on finite groups: held atCalifornia Institute of Technology, Proc. Sympos. Pure Math., VI, American Mathematical Society, pp. 101–105,ISBN 978-0821814062

References[1] http:/ / books. google. com/ books?id=Nb8rT4rm0EUC& pg=PA101

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Brauer–Suzuki–Wall theoremIn mathematics, the Brauer–Suzuki–Wall theorem, proved by Brauer, Suzuki & Wall (1958), characterizes theone-dimensional unimodular projective groups over finite fields.

ReferencesBrauer, R.; Suzuki, Michio; Wall, G. E. (1958), "A characterization of the one-dimensional unimodular projectivegroups over finite fields" [1], Illinois Journal of Mathematics 2: 718–745, ISSN 0019-2082, MR0104734

References[1] http:/ / projecteuclid. org/ euclid. ijm/ 1255448336

Burnside theoremIn mathematics, Burnside's theorem in group theory states that if G is a finite group of order

where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelianfinite simple group has order divisible by three distinct primes. Furthermore, as a consequence of the Feit-Thompsontheorem, one of those can be chosen to be 2.

HistoryThe theorem was proved by William Burnside in the early years of the 20th century.Burnside's theorem has long been one of the best-known applications of representation theory to the theory of finitegroups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970.

Outline of Burnside's proof1. By induction, it suffices to prove that a finite simple group G whose order has the form for primes p and q

is cyclic. Suppose then that the order of G has this form, but G is not cyclic. Suppose for definiteness that b >0.2. Using the modified class equation, G has a non-identity conjugacy class of size prime to q. Hence G either has a

non-trivial center, or has a conjugacy class of size for some positive integer r. The first possibility is excludedsince G is assumed simple, but not cyclic. Hence there is a non-central element x of G such that the conjugacyclass of x has size .

3. Application of column orthogonality relations and other properties of group characters and algebraic integers leadto the existence of a non-trivial irreducible character of G such that .

4. The simplicity of G then implies that any non-trivial complex irreducible representation is faithful, and it followsthat x is in the center of G, a contradiction.

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References1. James, Gordon; and Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge

University Press. ISBN 0-521-00392-X. See chapter 31.2. Fraleigh, John B. (2002) A First Course in Abstract Algebra (7th ed.). Addison Wesley. ISBN 0-201-33596-4.

Cartan's theoremIn mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:

1. The theorem that for a Lie group G, any closed subgroup is a Lie subgroup.[1]

2. A theorem on highest weight vectors in the representation theory of a semisimple Lie group.3. The equivalence between the category of connected real Lie groups and finite dimensional real Lie algebrasis called usually (in the literature of the second half of 20th century) Cartan's or Cartan-Lie theorem as it isproved by Élie Cartan whereas S. Lie has proved earlier just the infinitesimal version (local solvability ofMaurer-Cartan equations (see Maurer-Cartan form) or the equivalence between the finite dimensional Liealgebras and the category of local Lie groups). Lie listed his results as 3 direct and 3 converse theorems, theinfinitesimal variant of Cartan's theorem was essentially his 3rd converse theorem, hence Serre has called it inan influential book, the "third Lie theorem", the name which is historically somewhat misleading, but moreoften used in the recent decade in the connection to many generalizations.

See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributedto Élie and Henri Cartan.

Notes[1] See §26 of Cartan's article La théorie des groups finis et continus et l'Analysis Situs.

References• Cartan, Élie (1930), "La théorie des groupes finis et continus et l'Analysis Situs", Mémorial Sc. Math. XLII: 1–61• Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in

Mathematics, 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR1834454

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Cartan–Dieudonné theoremIn mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, is a theorem on thestructure of the automorphism group of symmetric bilinear spaces.

Statement of the theoremLet (V, b) be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to2. Then, every element of the orthogonal group O(V, b) is a composition of at most n reflections.

References• Sylvestre Gallot, Dominique Hulin, Jacques LaFontaine, Riemannian Geometry, Springer, 2004. ISBN

3540204938.• Jean H Gallier, Geometric Methods and Applications, Springer, 2000. ISBN 0387950443.

Cauchy's theorem (group theory)Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It statesthat if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then Gcontains an element of order p. That is, there is x in G so that p is the lowest non-zero number with xp = e, where e isthe identity element.The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group G dividesthe order of G. Cauchy's theorem implies that for any prime divisor p of the order of G, there is a subgroup of Gwhose order is p—the cyclic group generated by the element in Cauchy's theorem.Cauchy's theorem is generalised by Sylow's first theorem, which implies that if pn is any prime power dividing theorder of G, then G has a subgroup of order pn.

Statement and proofMany texts appear to prove the theorem with the use of strong induction and the class equation, though considerablyless machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof.Theorem: Let G be a finite group and p be a prime. If p divides the order of G, then G has an element of order p.Proof 1: We induct on n = |G| and consider the two cases where G is abelian or G is nonabelian. Suppose G isabelian. If G is simple, then it must be cyclic of prime order and trivially contains an element of order p. Otherwise,there exists a nontrivial, proper normal subgroup . If p divides |H|, then H contains an element of order p bythe inductive hypothesis, and thus G does as well. Otherwise, p must divide the index [G:H] by Lagrange's theorem,and we see the quotient group G/H contains an element of order p by the inductive hypothesis; that is, there exists anx in G such that (Hx)p = Hxp = H. Then there exists an element h1 in H such that h1xp = 1, the identity element of G.It is easily checked that for every element a in H there exists b in H such that bp = a, so there exists h2 in H so that h2p = h1. Thus h2x has order p, and the proof is finished for the abelian case.Suppose that G is nonabelian, so that its center Z is a proper subgroup. If p divides the order of the centralizer CG(a) for some noncentral element a (i.e. a is not in Z), then CG(a) is a proper subgroup and hence contains an element of order p by the inductive hypothesis. Otherwise, we must have p dividing the index [G:CG(a)], again by Lagrange's Theorem, for all noncentral a. Using the class equation, we have p dividing the left side of the equation (|G|) and also dividing all of the summands on the right, except for possibly |Z|. However, simple arithmetic shows p must also

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divide the order of Z, and thus the center contains an element of order p by the inductive hypothesis as it is a propersubgroup and hence of order strictly less than that of G. This completes the proof.Proof 2: This time we define the set of p-tuples whose elements are in the group G by

.Note that we can choose only (p-1) of the independently, since we are constrained by the product equal to theidentity. Thus , from which we deduce that p also divides

Define the action by , where is the cyclicgroup of order p.Then is the orbit of some element .The stabilizer is , from which we can deduce the order,

.

We have from the Orbit-Stabilizer Theorem that for each .

Take and the distinct orbits. Then .

Hence we know that .

p divides |X| implies that there is at least one other with the property that its orbit has order 1.

Then we have by the definition of X.

Since xj is in G this completes the proof.

UsesA practically immediate consequence of Cauchy's Theorem is a useful characterization of finite p-groups, where p isa prime. In particular, a finite group G is a p-group (i.e. all of its elements have order pk for some natural number k)if and only if G has order pn for some natural number n. It is also typical to use Cauchy's Theorem to prove the firstof Sylow's Theorems, though this is not required.

References• James McKay. Another proof of Cauchy's group theorem, American Math. Monthly, 66 (1959), p. 119.

External links• Cauchy's theorem [1] on PlanetMath• Proof of Cauchy's theorem [2] on PlanetMath

References[1] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=1569[2] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=2186

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Cayley's theoremIn group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to asubgroup of the symmetric group acting on G.[1] This can be understood as an example of the group action of G onthe elements of G.[2]

A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a groupunder function composition, called the symmetric group on G, and written as Sym(G).[3]

Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as(R,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are truefor groups in general.

HistoryAlthough Burnside[4] attributes the theorem to Jordan,[5] Eric Nummela[6] nonetheless argues that the standardname—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,[7] showed that thecorrespondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus anisomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at thetime, thus predating Jordan by 16 years or so.

Proof of the theoremWhere g is any element of G, consider the function fg : G → G, defined by fg(x) = g*x. By the existence of inverses,this function has a two-sided inverse, . So multiplication by g acts as a bijective function. Thus, fg is apermutation of G, and so is a member of Sym(G).The set is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish thisis to consider the function T : G → Sym(G) with T(g) = fg for every g in G. T is a group homomorphism because(using "•" for composition in Sym(G)):

for all x in G, and hence:

The homomorphism T is also injective since T(g) = idG (the identity element of Sym(G)) implies that g*x = x for all xin G, and taking x to be the identity element e of G yields g = g*e = e. Alternatively, T is also injective since, ifg*x=g'*x implies g=g' (by post-multiplying with the inverse of x, which exists because G is a group).Thus G is isomorphic to the image of T, which is the subgroup K.T is sometimes called the regular representation of G.

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Alternative setting of proofAn alternative setting uses the language of group actions. We consider the group as a G-set, which can be shownto have permutation representation, say .

Firstly, suppose with . Then the group action is by classification of G-orbits (alsoknown as the orbit-stabilizer theorem).Now, the representation is faithful if is injective, that is, if the kernel of is trivial. Suppose ∈ ker Then,

by the equivalence of the permutation representation and the group action. But since ∈ ker, and thus ker is trivial. Then im and thus the result follows by use of the first

isomorphism theorem.

Remarks on the regular group representationThe identity group element corresponds to the identity permutation. All other group elements correspond to apermutation that does not leave any element unchanged. Since this also applies for powers of a group element, lowerthan the order of that element, each element corresponds to a permutation which consists of cycles which are of thesame length: this length is the order of that element. The elements in each cycle form a left coset of the subgroupgenerated by the element.

Examples of the regular group representationZ2 = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 topermutation (12).Z3 = {0,1,2} with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 topermutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123)=(132).Z4 = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).The elements of Klein four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).S3 (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6group elements:

* e a b c d f permutation

e e a b c d f e

a a e d f b c (12)(35)(46)

b b f e d c a (13)(26)(45)

c c d f e a b (14)(25)(36)

d d c a b f e (156)(243)

f f b c a e d (165)(234)

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Notes[1] Jacobson (2009), p. 38.[2] Jacobson (2009), p. 72, ex. 1.[3] Jacobson (2009), p. 31.[4] Burnside, William (1911), Theory of Groups of Finite Order (2 ed.), Cambridge, ISBN 0486495752[5] Jordan, Camille (1870), Traite des substitutions et des equations algebriques, Paris: Gauther-Villars[6] Nummela, Eric (1980), "Cayley's Theorem for Topological Groups", American Mathematical Monthly (Mathematical Association of

America) 87 (3): 202–203, doi:10.2307/2321608, JSTOR 2321608[7] Cayley, Arthur (1854), "On the theory of groups as depending on the symbolic equation θn=1", Phil. Mag. 7 (4): 40–47

References• Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN 978-0-486-47189-1.

Cayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and WilliamHamilton) states that every square matrix over a commutative ring (including the real or complex field) satisfies itsown characteristic equation.More precisely:If A is a given n×n matrix and In  is the n×n identity matrix, then the characteristic polynomial of A is defined as

where "det" is the determinant operation. Since the entries of the matrix are (linear or constant) polynomials in λ, thedeterminant is also a polynomial in λ. The Cayley–Hamilton theorem states that "substituting" the matrix A for λ inthis polynomial results in the zero matrix:

The powers of λ that have become powers of A by the substitution should be computed by repeated matrixmultiplication, and the constant term should be multiplied by the identity matrix (the zeroth power of A) so that it canbe added to the other terms. The theorem allows An to be expressed as a linear combination of the lower matrixpowers of A.The Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix dividesits characteristic polynomial.

ExampleAs a concrete example, let

.

Its characteristic polynomial is given by

The Cayley–Hamilton theorem claims that, if we define

then

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which one can verify easily.

Illustration for specific dimensions and practical applicationsFor a 1×1 matrix A = (a), the characteristic polynomial is given by p(λ)=λ−a, and so p(A)=(a)−a(1)=(0) is obvious.For a 2×2 matrix,

the characteristic polynomial is given by p(λ)=λ2−(a+d)λ+(ad−bc), so the Cayley–Hamilton theorem states that

which is indeed always the case, evident by working out the entries of A2.For a general n×n invertible matrix A, i.e., one with nonzero determinant, A−1 can thus be written as an (n−1)-thorder polynomial expression in A: As indicated, the Cayley–Hamilton theorem amounts to the identity

with cn−1=−tr(A), etc., where tr(A) is the trace of the matrix A. This can then be written as

and, by multiplying both sides by , one is led to the compact expression for the inverse

For larger matrices, the expressions for the coefficients ck of the characteristic polynomial in terms of the matrixcomponents become increasingly complicated; but they can also be expressed in terms of traces of powers of thematrix A, using Newton's identities, thus resulting in more compact expressions (but which involve divisions bycertain integers).For instance, the coefficient −c1=a+d of λ above is just the trace of A, trA, while the constant coefficient c0=ad−bccan be written as ½((trA)2−tr(A2)). (Of course, it is also the determinant of A in this case.)In fact, this expression, ½((trA)2−tr(A2)), always gives the coefficient cn−2 of λn−2 in the characteristic polynomial ofany n×n matrix; so, for a 3×3 matrix A, the statement of the Cayley–Hamilton theorem can also be written as

where the right-hand side designates a 3×3 matrix with all entries reduced to zero.Similarly, one can write for a 4×4 matrix A:

and so on for larger matrices, with the increasingly complex expressions for the coefficients deducible fromNewton's identities.An alternate, practical method for obtaining these coefficients ck for a general n×n matrix, yielding the above onesvirtually by inspection, relies on

.Hence,

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where the exponential only needs be expanded to order λ−n, since p(λ) is of order n.The Cayley–Hamilton theorem always provides a relationship between the powers of A (though not always thesimplest one), which allows one to simplify expressions involving such powers, and evaluate them without having tocompute the power An or any higher powers of A. For instance the concrete 2×2 Example above can be written as

Then, for example, to calculate A4, observe

Proving the theorem in generalAs the examples above show, obtaining the statement of the Cayley–Hamilton theorem for an n×n matrix

requires two steps: first the coefficients ci of the characteristic polynomial are determined bydevelopment as a polynomial in t of the determinant

and then these coeffcients are used in a linear combination of powers of A that is equated to the n×n null matrix:

The left hand side can be worked out to an n×n matrix whose entries are (enormous) polynomial expressions in theset of entries of A, so the Cayley–Hamilton theorem states that each of these expressions are equivalent to 0.For any fixed value of n these identities can be obtained by tedious but completely straightforward algebraicmanipulations. None of these computations can show however why the Cayley–Hamilton theorem should be validfor matrices of all possible sizes n, so a uniform proof for all n is needed.

PreliminariesIf a vector v of size n happens to be an eigenvector of A with eigenvalue λ, in other words if , then

which is the null vector since (the eigenvalues of A are precisely the roots of p(t)). This holds for all possible eigenvalues λ, so the two matrices equated by the theorem certainly give the same (null) result when applied to any eigenvector. Now if A admits a basis of eigenvectors, in other words if A is diagonalizable, then the Cayley–Hamilton theorem must hold for A, since two matrices that give the same values when applied to each element of a basis must be equal. Not all matrices are diagonalizable, but for matrices with complex coefficients many of them are: the set of diagonalizable complex square matrices of a given size is dense in the set of all such square matrices (for a matrix to be diagonalizable it suffices for instance that its characteristic polynomial not have multiple roots). Now if any of the expressions that the theorem equates to 0 would not reduce to a null expression, in other words if it would be a nonzero polynomial in the coefficients of the matrix, then the set of complex matrices for which this expression happens to give 0 would not be dense in the set of all matrices, which would contradict the fact that the theorem holds for all diagonalizable matrices. Thus one can see that the

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Cayley–Hamilton theorem must be true.While this provides a valid proof, the argument is not very satisfactory, since the identities represented by thetheorem do not in any way depend on the nature of the matrix (diagonalizable or not), nor on the kind of entriesallowed (for matrices with real entries the diagonizable ones do not form a dense set, and it seems strange one wouldhave to consider complex matrices to see that the Cayley–Hamilton theorem holds for them). We shall therefore nowconsider only arguments that prove the theorem directly for any matrix using algebraic manipulations only; thesealso have the benefit of working for matrices with entries in any commutative ring.There is a great variety of such proofs of the Cayley–Hamilton theorem, of which several will be given here. Theyvary in the amount of abstract algebraic notions required to understand the proof. The simplest proofs use just thosenotions needed to formulate the theorem (matrices, polynomials with numeric entries, determinants), but involvetechnical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion. Itis possible to avoid such details, but at the price of involving more subtle algebraic notions: polynomials withcoefficients in a non-commutative ring, or matrices with unusual kinds of entries.

Adjugate matrices

All proofs below use the notion of the adjugate matrix of an n×n matrix M. This is a matrix whosecoefficients are given by polynomial expressions in the coefficients of M (in fact by certain (n − 1)×(n − 1)determinants), in such a way that one has the following fundamental relations

These relations are a direct consequence of the basic properties of determinants: evaluation of the (i,j) entry of thematrix product on the left gives the expansion by column j of the determinant of the matrix obtained from M byreplacing column i by a copy of column j, which is if and zero otherwise; the matrix product onthe right is similar, but for expansions by rows. Being a consequence of just algebraic expression manipulation, theserelations are valid for matrices with entries in any commutative ring (commutativity must be assumed fordeterminants to be defined in the first place). This is important to note here, because these relations will be appliedfor matrices with non-numeric entries such as polynomials.

A direct algebraic proofThis proof uses just the kind of objects needed to formulate the Cayley–Hamilton theorem: matrices withpolynomials as entries. The matrix whose determinant is the characteristic polynomial of A is such amatrix, and since polynomials form a commutative ring, it has an adjugate

Then according to the right hand fundamental relation of the adjugate one has

Since B is also a matrix with polynomials in t as entries, one can for each i collect the coefficients of in each entryto form a matrix Bi of numbers, such that one has

(the way the entries of B are defined makes clear that no powers higher than occur). While this looks like apolynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix withpolynomial entries as linear combination of constant matrices, and the coefficient has been written to the left ofthe matrix to stress this point of view. Now one can expand the matrix product in our equation by bilinearity

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Writing , one obtains an equality of two matrices withpolynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such anequality can hold only if in any matrix position the entry that is multiplied by a given power is the same on bothsides; it follows that the constant matrices with coefficient in both expressions must be equal. Writing theseequations for i from n down to 0 one finds

We multiply the equation of the coefficients of ti from the left by Ai, and sum up; the left-hand sides form atelescoping sum and cancel completely, which results in the equation

This completes the proof.

A proof using polynomials with matrix coefficientsThis proof is similar to the first one, but tries to give meaning to the notion of polynomial with matrix coefficientsthat was suggested by the expressions occurring in that proof. This requires considerable care, since it is somewhatunusual to consider polynomials with coefficients in a non-commutative ring, and not all reasoning that is valid forcommutative polynomials can be appied in this setting. Notably, while arithmetic of polynomials over acommutative ring models the arithmetic of polynomial functions, this is not the case over a non-commutative ring (infact there is no obvious notion of polynomial function in this case that is closed under multiplication). So whenconsidering polynomials in t with matrix coefficients, the variable t must not be thought of as an "unknown", but as aformal symbol that is to manipulated according to given rules; in particular one cannot just set t to a specific value.Let M = Mn(R) be the ring of n × n matrices with entries in some ring R (such as the real or complex numbers) thathas A as an element. Matrices with as coefficients polynomials in t, such as or its adjugate B in the firstproof, are elements of Mn(R[t]). By collecting like powers of t, such matrices can be written as "polynomials" in twith constant matrices as coefficients; write M[t] for the set of such polynomials. Since this set is in bijection withMn(R[t]), one defines arithmetic operations on it correspondingly, in particular multiplication is given by

respecting the order of the coefficient matrices from the two operands; obviously this gives a non-commutativemultiplication. Thus the identity

from the first proof can be viewed as one involving a multiplication of elements in M[t].At this point, it is tempting to set t equal to the matrix A, which makes the first factor on the left equal to the null matrix, and the right hand side equal to p(A); however, this is not an allowed operation when coefficients do not

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commute. It is possible to define a "right-evaluation map" evA : M[t] → M, which replaces each ti by the matrixpower Ai of A, where one stipulates that the power is always to be multiplied on the right to the correspondingcoefficient. However this map is not a ring homomorphism: the right-evaluation of a product differs in general fromthe product of the right-evaluations. This is so because multiplication of polynomials with matrix coefficients doesnot model multiplication of expressions containing unknowns: a product is defined assuming that t commutes withN, but this may fail if t is replaced by the matrix A.One can work around this difficulty in the particular situation at hand, since the above right-evaluation map doesbecome a ring homomorphism if the matrix A is in the center of the ring of coefficients, so that it commutes with allthe coefficients of the polynomials (the argument proving this is straightforward, exactly because commuting t withcoefficients is now justified after evaluation). Now A is not always in the center of M, but we may replace M with asmaller ring provided it contains all the coefficients of the polynomials in question: , A, and the coefficients of the polynomial B. The obvious choice for such a subring is the centralizer Z of A, the subring of all matrices thatcommute with A; by definition A is in the center of Z. This centralizer obviously contains , and A, but one has toshow that it contains the matrices . To do this one combines the two fundamental relations for adjugates, writingout the adjugate B as a polynomial:

Equating the coefficients shows that for each i, we have A Bi = Bi A as desired. Having found the proper setting inwhich evA is indeed a homomorphism of rings, one can complete the proof as suggested above:

This completes the proof.

A synthesis of the first two proofsIn the first proof, one was able to determine the coefficients Bi of B based on the right hand fundamental relation forthe adjugate only. In fact the first n equations derived can be interpreted as determining the quotient B of theEuclidean division of the polynomial on the left by the monic polynomial , while the finalequation expresses the fact that the remainder is zero. This division is performed in the ring of polynomials withmatrix coefficients. Indeed, even over a non-commutative ring, Euclidean division by a monic polynomial P isdefined, and always produces a unique quotient and remainder with the same degree condition as in the commutativecase, provided it is specified at which side one wishes P to be a factor (here that is to the left). To see that quotientand remainder are unique (which is the important part of the statement here), it suffices to write

as and observe that since P is monic, cannot havea degree less than that of P, unless .But the dividend and divisor used here both lie in the subring (R[A])[t], where R[A] is the subringof the matrix ring M generated by A: the R-linear span of all powers of A. Therefore the Euclidean division can infact be performed within that commutative polynomial ring, and of course it then gives the same quotient B andremainder 0 as in the larger ring; in particular this shows that B in fact lies in . But in this commutativesetting it is valid to set t to A in the equation , in other words apply the evaluation map

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which is a ring homomorphism, giving

just like in the second proof, as desired.In addition to proving the theorem, the above argument tells us that the coefficients of B are polynomials in A,while from the second proof we only knew that they lie in the centralizer Z of A; in general Z is a larger subring thanR[A], and not necessarily commutative. In particular the constant term lies in R[A]. Since A is anarbitrary square matrix, this proves that can always be expressed as a polynomial in (with coefficientsthat depend on ), something that is not obvious from the definition of the adjugate matrix. In fact the equationsfound in the first proof allow successively expressing , ..., , as polynomials in A, which leads to theidentity

valid for all n×n matrices, where is the characteristic polynomial of A. Notethat this identity implies the statement of the Cayley–Hamilton theorem: one may move to the right handside, multiply the resulting equation (on the left or on the right) by , and use the fact that

A proof using matrices of endomorphisms

As was mentioned above, the matrix in statement of the theorem is obtained by first evaluating thedeterminant and then substituting the matrix A for t; doing that subtitution into the matrix beforeevaluating the determinant is not meaningful. Nevertheless, it is possible to give an interpretation where isobtained directly as the value of a certain deteminant, but this requires a more complicated setting, one of matricesover a ring in which one can interpret both the entries of A, and all of A itself. One could take for this the ringM of n × n matrices over R, where the entry is realised as , and A as itself. But considering matriceswith matrices as entries might cause confusion with block matrices, which is not intended, as that gives the wrongnotion of determinant. It is clearer to distinguish A from the endomorphism φ of an n-dimensional vector space V (orfree R-module if R is not a field) defined by it in a basis e1, ..., en, and to take matrices over the ring End(V) of allsuch endomorphisms. Then is a possible matrix entry, while A designates the element of

whose entry is endomorphism of scalar multiplication by ; similarly In will beinterpreted as element of . However, since End(V) is not a commutative ring, no deteminant isdefined on ; this can only be done for matrices over a commutative subring of End(V). Now theentries of the matrix all lie in the subring R[φ] generated by the identity and φ, which is commutative.Then a determinant map is defined, and evaluates to the value p(φ) of thecharacteristic polynomial of A at φ (this holds independently of the relation between A and φ); the Cayley–Hamiltontheorem states that p(φ) is the null endomorphism.In this form, the following proof can be obtained from that of (Atiyah & MacDonald 1969, Prop. 2.4) (which in factis the more general statement related to the Nakayama lemma; one takes for the ideal in that proposition the wholering R). The fact that A is the matrix of φ in the basis e1, ..., enmeans that

One can interpret these as n components of one equation in Vn, whose members can be written using thematrix-vector product that is defined as usual, but with individual entries

and being "multiplied" by forming ; this gives:

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where is the element whose component i is ei (in other words it is the basis e1, ..., en of V written as acolumn of vectors). Writing this equation as

one recognizes the transpose of the matrix considered above, and its determinant (as element of) is also p(φ). To derive from this equation that , one left-multiplies by the

adjugate matrix of , which is defined in the matrix ring , giving

the associativity of matrix-matrix and matrix-vector multiplication used in the first step is a purely formal property ofthose operations, independent of the nature of the entries. Now component i of this equation says that

; thus p(φ) vanishes on all ei, and since these elements generate V it follows that, completing the proof.

One additional fact that follows from this proof is that the matrix A whose characteristic polynomial is taken neednot be identical to the value φ substituted into that polynomial; it suffices that φ be an endomorphism of V satisfyingthe initial equations φ(ei) = Σj Aj,iej for some sequence of elements e1,...,en that generate V (which space might havesmaller dimension than n, or in case the ring R is not a field it might not be a free module at all).

A bogus "proof": p(A) = det(AIn − A) = det(A − A) = 0One elementary but incorrect argument for the theorem is to "simply" take the definition

and substitute for , obtaining

There are many ways to see why this argument is wrong. First, in Cayley–Hamilton theorem, p(A) is an n×n matrix.However, the right hand side of the above equation is the value of a determinant, which is a scalar. So they cannotbe equated unless n = 1 (i.e. A is just a scalar). Second, in the expression , the variable actuallyoccurs at the diagonal entries of the matrix . To illustrate, consider the characteristic polynomial in theprevious example again:

If one substitutes the entire matrix for in those positions, one obtains

in which the "matrix" expression is simply not a valid one. Note, however, that if scalar multiples of identitymatrices instead of scalars are subtracted in the above, i.e. if the substitution is performed as

then the determinant is indeed zero, but the expanded matrix in question does not evaluate to ; nor can its determinant (a scalar) be compared to (a matrix). So the argument that

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still does not apply.Actually, if such an argument holds, it should also hold when other multilinear forms instead of determinant is used.For instance, if we consider the permanent function and define , then by the sameargument, we should be able to "prove" that q(A) = 0. But this statement is demonstrably wrong. In the2-dimensional case, for instance, the permanent of a matrix is given by

So, for the matrix in the previous example,

Yet one can verify that One of the proofs for Cayley–Hamilton theorem above bears some similarity to the argument that

. By introducing a matrix with non-numeric coefficients, one can actually let lives inside a matrix entry, but then is not equal to , and the conclusion is reached differently.

Abstraction and generalizationsThe above proofs show that the Cayley–Hamilton theorem holds for matrices with entries in any commutative ringR, and that p(φ) = 0 will hold whenever φ is an endomorphism of an R module generated by elements e1,...,en thatsatisfies for j = 1,...,n. This more general version of the theorem is the source of the celebrated

Nakayama lemma in commutative algebra and algebraic geometry.

References• Atiyah, M. F.; MacDonald, I. G. (1969), Introduction to Commutative Algebra, Westview Press,

ISBN 0-201-40751-5

External links• A proof from PlanetMath. [1]

• The Cayley-Hamilton Theorem [2] at MathPages• T. Kaczorek (2001), "Cayley–Hamilton theorem" [3], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,

Springer, ISBN 978-1556080104

References[1] http:/ / planetmath. org/ ?op=getobj& from=objects& id=7308[2] http:/ / www. mathpages. com/ home/ kmath640/ kmath640. htm[3] http:/ / eom. springer. de/ C/ c022410. htm

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ChevalleyShephardTodd theorem 57

Chevalley–Shephard–Todd theoremIn mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring ofinvariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group isgenerated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was firstproved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soonafterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in thenon-modular case by Jean-Pierre Serre.

Statement of the theoremLet V be a finite-dimensional vector space over a field K and let G be a finite subgroup of the general linear groupGL(V). An element s of GL(V) is called a pseudoreflection if it fixes a codimension one subspace of V and is not theidentity transformation I, or equivalently, if the kernel Ker (s − I) has codimension one in V. Assume that the orderof G is relatively prime to the characteristic of K (the so-called non-modular case). Then the following threeproperties are equivalent:• The group G is generated by pseudoreflections.• The algebra of invariants K[V]G is a (free) polynomial algebra.• The algebra K[V] is a free module over K[V]G.In the case when the field K is the field C of complex numbers, the first condition is usually stated as "G is acomplex reflection group". Shephard and Todd derived a full classification of such groups.

Examples• Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicative group

of the field K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n, where n isits order, so G is generated by pseudoreflections. In this case, K[V] = K[x] is the polynomial ring in one variableand the algebra of invariants of G is the subalgebra generated by xn, hence it is a polynomial algebra.

• Let V = Kn be the standard n-dimensional vector space and G be the symmetric group Sn acting by permutations ofthe elements of the standard basis. The symmetric group is generated by transpositions (ij), which act byreflections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants is thepolynomial algebra generated by the elementary symmetric functions e1, … en.

• Let V = K2 and G be the cyclic group of order 2 acting by ±I. In this case, G is not generated by pseudoreflections,since the nonidentity element s of G acts without fixed points, so that dim Ker (s − I) = 0. On the other hand, thealgebra of invariants is the subalgebra of K[V] = K[x, y] generated by the homogeneous elements x2, xy, and y2 ofdegree 2. This subalgebra is not a polynomial algebra because of the relation x2y2 = (xy)2.

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GeneralizationsBroer (2007) gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic.There has been much work on the question of when a reductive algebraic group acting on a vector space has apolynomial ring of invariants. In the case when the algebraic group is simple and the representation is irreducible allcases when the invariant ring is polynomial have been classified by Schwarz (1978)In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so itis a finite rank free module over a polynomial subring.

References• Broer, Abraham (2007), On Chevalley-Shephard-Todd's theorem in positive characteristic, [], arXiv:0709.0715• Chevalley, Claude (1955), "Invariants of finite groups generated by reflections", Amer. J. Of Math. 77 (4):

778–782, doi:10.2307/2372597, JSTOR 2372597• Neusel, Mara D.; Smith, Larry (2002), Invariant Theory of Finite Groups, American Mathematical Society,

ISBN 0-8218-2916-5• Shephard, G. C.; Todd, J. A. (1954), "Finite unitary reflection groups", Canadian J. Math. 6: 274–304,

doi:10.4153/CJM-1954-028-3• Schwarz, G. (1978), "Representations of simple Lie groups with regular rings of invariants", Invent. Math. 49 (2):

167–191, doi:10.1007/BF01403085• Smith, Larry (1997), "Polynomial invariants of finite groups. A survey of recent developments" [1], Bull. Amer.

Math. Soc. 34 (3): 211–250, doi:10.1090/S0273-0979-97-00724-6, MR1433171• Springer, T. A. (1977), Invariant Theory, Springer, ISBN 0-387-08242-5

References[1] http:/ / www. ams. org/ bull/ 1997-34-03/ S0273-0979-97-00724-6/

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ChevalleyWarning theorem 59

Chevalley–Warning theoremIn algebra, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently manyvariables over a finite field have solutions. It was proved by Ewald Warning (1936) and a slightly weaker form of thetheorem, known as Chevalley's theorem, was proved by Chevalley (1936). Chevalley's theorem implied Artin's andDickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x).

Statement of the theoremsConsider a system of polynomial equations

where the are polynomials with coefficients in a finite field and such that the number of variables satisfies

where is the total degree of . The Chevalley–Warning theorem states that the number of common solutionsis divisible by the characteristic of . Chevalley's theorem states that if the system has the

trivial solution , i.e. if the polynomials have no constant terms, then the system also has anon-trivial solution .Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since is at least 2.

Both theorems are best possible in the sense that, given any , the list has total degreeand only the trivial solution. Alternatively, using just one polynomial, we can take P1 to be the degree n

polynomial given by the norm of x1a1 + ... + xnan where the elements a form a basis of the finite field of order pn.

Proof of Warning's theoremIf i<p−1 then

so the sum over Fn of any polynomial in x1,...,xn of degree less than n(p−1) also vanishes.The total number of common solutions mod p of P1 = ... = Pr = 0 is

because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials Pi is less than nthen this vanishes by the remark above.

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Artin's conjectureIt is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjecturedby Emil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closedfields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn'stheorem.

The Ax–Katz theoremThe Ax–Katz theorem, named after James Ax and Nicholas Katz, determines more accurately a power of thecardinality of dividing the number of solutions; here, if is the largest of the , then the exponent canbe taken as the ceiling function of

The Ax–Katz result has an interpretation in étale cohomology as a divisibility result for the (reciprocals of) thezeroes and poles of the local zeta-function. Namely, the same power of divides each of these algebraic integers.

References• Artin, Emil (1982), Lang, Serge.; Tate, John, eds., Collected papers, Berlin, New York: Springer-Verlag,

ISBN 978-0-387-90686-7, MR671416• Ax, James (1964), "Zeros of polynomials over finite fields", American Journal of Mathematics 86: 255–261,

doi:10.2307/2373163, MR0160775• Chevalley, Claude (1936), "Démonstration d'une hypothèse de M. Artin" (in French), Abhandlungen aus dem

Mathematischen Seminar der Universität Hamburg 11: 73–75, doi:10.1007/BF02940714, JFM 61.1043.01,Zbl 0011.14504

• Katz, Nicholas M. (1971), "On a theorem of Ax", Amer. J. Math. 93 (2): 485–499, doi:10.2307/2373389• Warning, Ewald (1936), "Bemerkung zur vorstehenden Arbeit von Herrn Chevalley" (in German), Abhandlungen

aus dem Mathematischen Seminar der Universität Hamburg 11: 76–83, doi:10.1007/BF02940715,JFM 61.1043.02, Zbl 0011.14601

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Classification of finite simple groups 61

Classification of finite simple groupsIn mathematics, the classification of the finite simple groups is a theorem stating that every finite simple groupbelongs to one of four categories described below. These groups can be seen as the basic building blocks of all finitegroups, in much the same way as the prime numbers are the basic building blocks of the natural numbers. TheJordan–Hölder theorem is a more precise way of stating this fact about finite groups.The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about100 authors, published mostly between 1955 and 2004. Gorenstein, Lyons, and Solomon are gradually publishing asimplified and revised version of the proof.

Statement of the classification theoremTheorem. Every finite simple group is isomorphic to one of the following groups:• A cyclic group with prime order;• An alternating group of degree at least 5;• A simple group of Lie type, including both

• the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonaltransformations over a finite field;

• the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lietype).

• The 26 sporadic simple groups.The classification theorem has applications in many branches of mathematics, as questions about the structure offinite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finitesimple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking eachfamily of simple groups and each sporadic group.Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature ashe had been misinformed about the proof of the classification of quasithin groups. The completed proof of theclassification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221 page proof for themissing quasithin case.

Overview of the proof of the classification theoremGorenstein (1982, 1983) wrote two volumes outlining the low rank and odd characteristic part of the proof, andMichael Aschbacher, Richard Lyons, and Stephen D. Smith et al. (2011) wrote a 3rd volume covering the remainingcharacteristic 2 case. The proof can be broken up into several major pieces as follows:

Groups of small 2-rankThe simple groups of low 2-rank are mostly groups of Lie type of small rank over fields of odd characteristic,together with five alternating and seven characteristic 2 type and nine sporadic groups.The simple groups of small 2-rank include:• Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit-Thompson theorem.• Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or

generalized quaternion, which are handled with the Brauer-Suzuki theorem: in particular there are no simplegroups of 2-rank 1.

• Groups of 2-rank 2. Alperin showed that the Sylow subgoup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done by the Gorenstein–Walter theorem which showed that the

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only simple groups are isomorphic to L2(q) for q odd or A7, the second and third cases were done by theAlperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L3(q) or U3(q)for q odd or M11, and the last case was done by Lyons who showed that U3(4) is the only simple possibility.

• Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theorem.The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modularcharacter theory, which is almost never directly used elsewhere in the classification.All groups not of small 2 rank can be split into two major classes: groups of component type and groups ofcharacteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that itsSylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaksdown, because theorems such as the signalizer functor theorem only work for groups with elementary abeliansubgroups of rank at least 3.)

Groups of component typeA group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component (whereO(C) is the core of C, the maximal normal subgroup of odd order). These are more or less the groups of Lie type ofodd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in thiscase is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem, which statesthat every component of C/O(C) is the image of a component of C.The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimplegroup, which can be assumed to be already known by induction. So to classify these groups one takes every centralextension of every known finite simple group, and finds all simple groups with a centralizer of involution with thisas a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groupsand 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or oversmall fields behave differently from the general case and have to be treated separately, and the groups of Lie type ofeven and odd characteristic are also quite different.

Groups of characteristic 2 typeA group is of characteristic 2 type if the generalized Fitting subgroup F*(Y) of every 2-local subgroup Y is a 2-group.As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of othersthat are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rankcases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which isoften (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type incharacteristic 2.The rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notorious quasithingroups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 overfields of characteristic 2.Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem, proved by Aschbacher forrank 3 and by Gorenstein and Lyons for rank at least 4. The three classes are groups of GF(2) type (classified mainlyby Timmesfeld), groups of "standard type" for some odd prime (classified by the Gilman-Griess theorem and workby several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simplegroups. The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rankat least 3 or 4.

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Existence and uniqueness of the simple groupsThe main part of the classification produces a characterization of each simple group. It is then necessary to checkthat there exists a simple group for each characterization and that it is unique. This gives a large number of separateproblems; for example, the original proofs of existence and uniqueness of the monster totaled about 200 pages, andthe identification of the Ree groups by Thompson and Bombieri was one of the hardest parts of the classification.Many of the existence proofs and some of the uniqueness proofs for the sporadic proofs originally used computercalculations, some of which have since been replaced by shorter hand proofs.

History of the proof

Gorenstein's programIn 1972 Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups,consisting of the following 16 steps:1. Groups of low 2-rank. This was essentially done by Gorenstein and Harada, who classified the groups with

sectional 2-rank at most 4. Most of the cases of 2-rank at most 2 had been done by the time Gorenstein announcedhis program.

2. The semisimplicity of 2-layers. The problem is to prove that the 2-layer of the centralizer of an involution in asimple group is semisimple.

3. Standard form in odd characteristic. If a group has an involution with a 2-component that is a group of Lie type ofodd characteristic, the goal is to show that it has a centralizer of involution in "standard form" meaning that acentralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of2-rank 1.

4. Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in"standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classicalinvolution theorem.

5. Quasi-standard form6. Central involutions7. Classification of alternating groups. More precisely, show that if a simple group has8. Some sporadic groups9. Thin groups. The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified

by Aschbacher in 197810. Groups with a strongly p-embedded subgroup for p odd11. The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for

nonsolvable signalizer functors. This was solved by McBride in 1982.12. Groups of characteristic p type. This is the problem of groups with a strongly p-embedded 2-local subgroup with

p odd, which was handled by Aschbacher.13. Quasithin groups. A quasithin group is one whose 2-local subgroups have p-rank at most 2 for all odd primes p,

and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher andSmith in 2004.

14. Groups of low 2-local 3-rank. This was essentially solved by Aschbacher's trichotomy theorem for groups withe(G)=3. The main change is that 2-local 3-rank is replaced by 2-local p-rank for odd primes.

15. Centralizers of 3-elements in standard form. This was essentially done by the Trichotomy theorem.16. Classification of simple groups of characteristic 2 type. This was handled by the Gilman-Griess theorem, with

3-elements replaced by p-elements for odd primes.

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Timeline of the proofMany of the items in the list below are taken from Solomon (2001). The date given is usually the publication date ofthe complete proof of a result, which is sometimes several years later than the proof or first announcement of theresult, so some of the items appear in the "wrong" order.

Publicationdate

1832 Galois introduces normal subgroups and finds the simple groups An (n≥5) and PSL2(Fp) (p≥5)

1854 Cayley defines abstract groups

1861 Mathieu finds the first two Mathieu groups M11, M12, the first sporadic simple groups.

1870 Jordan lists some simple groups: the alternating and projective special linear ones, and emphasizes the importance of the simplegroups.

1872 Sylow proves the Sylow theorems

1873 Mathieu finds three more Mathieu groups M22, M23, M24.

1892 Otto Hölder proves that the order of any nonabelian finite simple group must be a product of at least 4 primes, and asks for aclassification of finite simple groups.

1893 Cole classifies simple groups of order up to 660

1896 Frobenius and Burnside begin the study of character theory of finite groups.

1899 Burnside classifies the simple groups such that the centralizer of every involution is a non-trivial elementary abelian 2-group.

1901 Frobenius proves that a Frobenius group has a Frobenius kernel, so in particular is not simple.

1901 Dickson defines classical groups over arbitrary finite fields, and exceptional groups of type G2 over fields of odd characteristic.

1901 Dickson introduces the exceptional finite simple groups of type E6.

1904 Burnside uses character theory to prove Burnside's theorem that the order of any non-abelian finite simple group must be divisibleby at least 3 distinct primes.

1905 Dickson introduces simple groups of type G2 over fields of even characteristic

1911 Burnside conjectures that every non-abelian finite simple group has even order

1928 Hall proves the existence of Hall subgroups of solvable groups

1933 Hall begins his study of p-groups

1935 Brauer begins the study of modular characters.

1936 Zassenhaus classifies finite sharply 3-transitive permutation groups

1938 Fitting introduces the Fitting subgroup and proves Fitting's theorem that for solvable groups the Fitting subgroup contains itscentralizer.

1942 Brauer describes the modular characters of a group divisible by a prime to the first power.

1954 Brauer classifies simple groups with GL2(Fq) as the centralizer of an involution.

1955 The Brauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite,suggesting an attack on the classification using centralizers of involutions.

1955 Chevalley introduces the Chevalley groups, in particular introducing exceptional simple groups of types F4, E7, and E8.

1956 Hall–Higman theorem

1957 Suzuki shows that all finite simple CA groups of odd order are cyclic.

1958 The Brauer–Suzuki–Wall theorem characterizes the projective special linear groups of rank 1, and classifies the simple CA groups.

1959 Steinberg introduces the Steinberg groups, giving some new finite simple groups, or types 3D4 and 2E6 (the latter wereindependently found at about the same time by Jacques Tits).

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1959 The Brauer–Suzuki theorem about groups with generalized quaternion Sylow 2-subgroups shows in particular that none of them aresimple.

1960 Thompson proves that a group with a fixed-point-free automorphism of prime order is nilpotent.

1960 Feit, Hall, and Thompson show that all finite simple CN groups of odd order are cyclic.

1960 Suzuki introduces the Suzuki groups, with types 2B2.

1961 Ree introduces the Ree groups, with types 2F4 and 2G2.

1963 Feit and Thompson prove the odd order theorem.

1964 Tits introduces BN pairs for groups of Lie type and finds the Tits group

1965 The Gorenstein–Walter theorem classifies groups with a dihedral Sylow 2-subgroup.

1966 Glauberman proves the Z* theorem

1966 Janko introduces the Janko group J1, the first new sporadic group for about a century.

1968 Glauberman proves the ZJ theorem

1968 Higman and Sims introduce the Higman–Sims group

1968 Conway introduces the Conway groups

1969 Walter's theorem classifies groups with abelian Sylow 2-subgroups

1969 Introduction of the Suzuki sporadic group, the Janko group J2, the Janko group J3, the McLaughlin group, and the Held group.

1969 Gorenstein introduces signalizer functors based on Thompson's ideas.

1970 Bender introduced the generalized Fitting subgroup

1970 The Alperin–Brauer–Gorenstein theorem classifies groups with quasi-dihedral or wreathed Sylow 2-subgroups, completing theclassification of the simple groups of 2-rank at most 2

1971 Fischer introduces the three Fischer groups

1971 Thompson classifies quadratic pairs

1971 Bender classifies group with a strongly embedded subgroup

1972 Gorenstein proposes a 16-step program for classifying finite simple groups; the final classification follows his outline quite closely.

1972 Lyons introduces the Lyons group

1973 Rudvalis introduces the Rudvalis group

1973 Fisher discovers the baby monster group (unpublished), which Fischer and Griess use to discover the monster group, which in turnleads Thompson to the Thompson sporadic group and Norton to the Harada–Norton group (also found in a different way byHarada).

1974 Thompson classifies N-groups, groups all of whose local subgroups are solvable.

1974 The Gorenstein–Harada theorem classifies the simple groups of sectional 2-rank at most 4, dividing the remaining finite simplegroups into those of component type and those of characteristic 2 type.

1974 Tits shows that groups with BN pairs of rank at least 3 are groups of Lie type

1974 Aschbacher classifies the groups with a proper 2-generated core

1975 Gorenstein and Walter prove the L-balance theorem

1976 Glauberman proves the solvable signalizer functor theorem

1976 Aschbacher proves the component theorem, showing roughly that groups of odd type satisfying some conditions have a componentin standard form. The groups with a component of standard form were classified in a large collection of papers by many authors.

1976 O'Nan introduces the O'Nan group

1976 Janko introduces the Janko group J4, the last sporadic group to be discovered

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1977 Aschbacher characterizes the groups of Lie type of odd characteristic in his classical involution theorem. After this theorem, whichin some sense deals with "most" of the simple groups, it was generally felt that the end of the classification was in sight.

1978 Timmesfeld proves the O2 extraspecial theorem, breaking the classification of groups of GF(2)-type into several smaller problems.

1978 Aschbacher classifies the thin finite groups, which are mostly rank 1 groups of Lie type over fields of even characteristic.

1981 Bombieri uses elimination theory to complete Thompson's work on the characterization of Ree groups, one of the hardest steps ofthe classification.

1982 McBride proves the signalizer functor theorem for all finite groups.

1982 Griess constructs the monster group by hand

1983 The Gilman–Griess theorem classifies groups groups of characteristic 2 type and rank at least 4 with standard components, one ofthe three cases of the trichotomy theorem.

1983 Aschbacher proves that no finite group satisfies the hypothesis of the uniqueness case, one of the three cases given by thetrichotomy theorem for groups of characteristic 2 type.

1983 Gorenstein and Lyons prove the trichotomy theorem for groups of characteristic 2 type and rank at least 4, while Aschbacher doesthe case of rank 3. This divides these groups into 3 subcases: the uniqueness case, groups of GF(2) type, and groups with a standardcomponent.

1983 Gorenstein announces the proof of the classification is complete, somewhat prematurely as the proof of the quasithin case wasincomplete.

1994 Gorenstein, Lyons, and Solomon begin publication of the revised classification

2004 Aschbacher and Smith publish their work on quasithin groups (which are mostly groups of Lie type of rank at most 2 over fields ofeven characteristic), filling the last (known) gap in the classification.

Second-generation classificationThe proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme lengthof the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generationclassification proof. This effort, called "revisionism", was originally led by Daniel Gorenstein.As of 2005, six volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994,1996, 1998, 1999, 2002, 2005), with most of the balance of the proof in manuscript. It is estimated that the newproof will eventually fill approximately 5,000 pages. (This length stems in part from second generation proof beingwritten in a more relaxed style.) Aschbacher and Smith wrote their two volumes devoted to the quasithin case insuch a way that those volumes can be part of the second generation proof.Gorenstein and his collaborators have given several reasons why a simpler proof is possible.• The most important is that the correct, final statement of the theorem is now known. Simpler techniques can be

applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those whoworked on the first generation proof did not know how many sporadic groups there were, and in fact some of thesporadic groups (e.g., the Janko groups) were discovered while proving other cases of the classification theorem.As a result, many of the pieces of the theorem were proved using techniques that were overly general.

• Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealingwith important special cases. Much of the work of proving these theorems was devoted to the analysis ofnumerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can bepostponed until the most powerful assumptions can be applied. The price paid under this revised strategy is thatthese first generation theorems no longer have comparatively short proofs, but instead rely on the completeclassification.

• Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, familiesand subfamiles of finite simple groups were identified multiple times. The revised proof eliminates these

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redundancies by relying on a different subdivision of cases.• Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal.Aschbacher (2004) has called the work on the classification problem by Ulrich Meierfrankenfeld, BerndStellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups incharacteristic 2 uniformly using the amalgam method.

References• Aschbacher, Michael (2004). "The Status of the Classification of the Finite Simple Groups" (http:/ / www. ams.

org/ notices/ 200407/ fea-aschbacher. pdf). Notices of the American Mathematical Society.• Aschbacher, Michael; Lyons, Richard; Smith, Stephen D.; Solomon, Ronald (2011), The Classification of Finite

Simple Groups: Groups of Characteristic 2 Type (http:/ / www. ams. org/ bookstore?fn=20& ikey=SURV-172),Mathematical Surveys and Monographs, 172, ISBN 978-0-8218-5336-8

• Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott(1985), Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, OxfordUniversity Press, ISBN 0198531990

• Gorenstein, D. (1979), "The classification of finite simple groups. I. Simple groups and local analysis", AmericanMathematical Society. Bulletin. New Series 1 (1): 43–199, doi:10.1090/S0273-0979-1979-14551-8,ISSN 0002-9904, MR513750

• Gorenstein, D. (1982), Finite simple groups, University Series in Mathematics, New York: Plenum PublishingCorp., ISBN 978-0-306-40779-6, MR698782

• Gorenstein, D. (1983), The classification of finite simple groups. Vol. 1. Groups of noncharacteristic 2 type, TheUniversity Series in Mathematics, Plenum Press, ISBN 978-0-306-41305-6, MR746470

• Daniel Gorenstein (1985), "The Enormous Theorem", Scientific American, vol. 253, no. 6, pp. 104–115.• Gorenstein, D. (1986), "Classifying the finite simple groups", American Mathematical Society. Bulletin. New

Series 14 (1): 1–98, doi:10.1090/S0273-0979-1986-15392-9, ISSN 0002-9904, MR818060• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1994), The classification of the finite simple groups (http:/ /

www. ams. org/ online_bks/ surv401), Mathematical Surveys and Monographs, 40, Providence, R.I.: AmericanMathematical Society, ISBN 978-0-8218-0334-9, MR1303592

• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1996), The classification of the finite simple groups. Number2. Part I. Chapter G (http:/ / www. ams. org/ online_bks/ surv402), Mathematical Surveys and Monographs, 40,Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0390-5, MR1358135

• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1998), The classification of the finite simple groups. Number3. Part I. Chapter A, Mathematical Surveys and Monographs, 40, Providence, R.I.: American MathematicalSociety, ISBN 978-0-8218-0391-2, MR1490581

• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1999), The classification of the finite simple groups. Number4. Part II. Chapters 1–4, Mathematical Surveys and Monographs, 40, Providence, R.I.: American MathematicalSociety, ISBN 978-0-8218-1379-9, MR1675976

• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (2002), The classification of the finite simple groups. Number5. Part III. Chapters 1–6, Mathematical Surveys and Monographs, 40, Providence, R.I.: American MathematicalSociety, ISBN 978-0-8218-2776-5, MR1923000

• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (2005), The classification of the finite simple groups. Number6. Part IV, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society,ISBN 978-0-8218-2777-2, MR2104668

• Mark Ronan, Symmetry and the Monster, ISBN 978-0-19-280723-6, Oxford University Press, 2006. (Conciseintroduction for lay reader)

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• Marcus du Sautoy, Finding Moonshine, Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction forthe lay reader)

• Ron Solomon (1995) " On Finite Simple Groups and their Classification, (http:/ / www. ams. org/ notices/199502/ solomon. pdf)" Notices of the American Mathematical Society. (Not too technical and good on history)

• Solomon, Ronald (2001), "A brief history of the classification of the finite simple groups" (http:/ / www. ams.org/ bull/ 2001-38-03/ S0273-0979-01-00909-0/ S0273-0979-01-00909-0. pdf), American Mathematical Society.Bulletin. New Series 38 (3): 315–352, doi:10.1090/S0273-0979-01-00909-0, ISSN 0002-9904, MR1824893 –article won Levi L. Conant prize (http:/ / www. ams. org/ notices/ 200604/ comm-conant. pdf) for exposition

• Thompson, John G. (1984), "Finite nonsolvable groups", in Gruenberg, K. W.; Roseblade, J. E., Group theory.Essays for Philip Hall, Boston, MA: Academic Press, pp. 1–12, ISBN 978-0-12-304880-6, MR780566

• Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, 251, Berlin, New York:Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792

External links• ATLAS of Finite Group Representations. (http:/ / brauer. maths. qmul. ac. uk/ Atlas/ v3/ ) Searchable database of

representations and other data for many finite simple groups.• Elwes, Richard, " An enormous theorem: the classification of finite simple groups, (http:/ / plus. maths. org/

issue41/ features/ elwes/ index. html)" Plus Magazine, Issue 41, December 2006. For laypeople.• Madore, David (2003) Orders of nonabelian simple groups. (http:/ / www. eleves. ens. fr:8080/ home/ madore/

math/ simplegroups. html) Includes a list of all nonabelian simple groups up to order 1010.

Cohn's irreducibility criterionArthur Cohn's irreducibility criterion is a test to determine whether a polynomial is irreducible in .The criterion is often stated as follows:

If a prime number is expressed in base 10 as (where) then the polynomial

is irreducible in .The theorem can be generalized to other bases as follows:

Assume that is a natural number and is apolynomial such that . If is a prime number then is irreducible in .

The base-10 version of the theorem attributed to Cohn by Pólya and Szegő in one of their books[1] while thegeneralization to any base, 2 or greater, is due to Brillhart, Filaseta, and Odlyzko[2] .In 2002, Ram Murty gave a simplified proof as well as some history of the theorem in a paper that is availableonline.[3] .The converse of this criterion is that, if p is an irreducible polynomial with integer coefficients that have greatestcommon divisor 1, then there exists a base such that the coefficients of p form the representation of a prime numberin that base; this is the Bunyakovsky conjecture and its truth or falsity remains an open question.

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Historical notes• Polya and Szegő gave their own generalization but it has many side conditions (on the locations of the roots, for

instance) so it lacks the elegance of Brillhart's, Filaseta's, and Odlyzko's generalization.• It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn, a student of Issai Schur

who was awarded his PhD in Berlin in 1921.[4]

References[1] George Pólya; Gábor Szegő (1925). Aufgaben und Lehrsätze aus der Analysis, Bd 2. Springer, Berlin. OCLC 73165700. English translation

in: George Pólya; Gabor Szegö (2004). Problems and theorems in analysis, volume 2. 2. Springer. p. 137. ISBN 3-540-63686-2.[2] Brillhart, John; Michael Filaseta, Andrew Odlyzko (1981). "On an irreducibility theorem of A. Cohn". Canadian Journal of Mathematics 33

(5): 1055–1059. doi:10.4153/CJM-1981-080-0.[3] Murty, Ram (2002). "Prime Numbers and Irreducible Polynomials" (http:/ / www. mast. queensu. ca/ ~murty/ polya4. dvi). American

Mathematical Monthly (The American Mathematical Monthly, Vol. 109, No. 5) 109 (5): 452–458. doi:10.2307/2695645. JSTOR 2695645. .(dvi file)

[4] Arthur Cohn's entry at the Mathematics Genealogy Project (http:/ / genealogy. math. ndsu. nodak. edu/ html/ id. phtml?id=17963)

External links• A. Cohn's irreducibility criterion (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=6194) on

PlanetMath

Cramer's ruleIn linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linearequations with as many equations as unknowns, valid in those cases where there is a unique solution. The solution isexpressed in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it byreplacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer(1704–1752), who published the rule in his 1750 Introduction à l'analyse des lignes courbes algébriques(Introduction to the analysis of algebraic curves), although Colin Maclaurin also published the method in his 1748Treatise of Algebra (and probably knew of the method as early as 1729).[1] [2]

General caseConsider a system of linear equations represented in matrix multiplication form as follows:

where the square matrix is invertible and the vector is the column vector of the variables.Then the theorem states that:

where is the matrix formed by replacing the ith column of by the column vector .The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. Ithas recently been shown that Cramer's rule can be implemented in O(n3) time,[3] which is comparable to morecommon methods of solving systems of linear equations, such as Gaussian elimination.

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ProofThe proof for Cramer's rule is very simple, in fact it uses just two properties of determinants. The first property isthat adding any multiple of one column to another does not change the value of the determinant, while the secondproperty is that multiplying every element of one column by a factor will multiply the value of the determinant bythe same factor.Given n linear equations with n variables .

Cramer's rule gives, for the value of , the expression:

which can be verified using the aforementioned properties of determinants. In fact, substituting for from the equations of the system, this quotient is equal to

By subtracting from the first column the second multiplied by , the third column multiplied by , and so onuntil the last column multiplied by , it is found to be equal to

,

and according to the remaining property of determinants the common factor of in the first column of thenumerator can be factored out of the determinant. Therefore this is equal to

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.

In the same way, if the columns of b's is used to replace the k-th column of the matrix of the system of equations theresult will be equal to . As a result we get that

These expressions for can be put into matrix notation as follows. First do a Laplace expansion (aka cofactorexpansion) on the determinants which are in the numerators using the columns which contain , , , .Thus Cramer's rule becomes;

this is equivalent to the matrix equation

Where are the cofactors of the coefficient matrix [A]

is the determinant of the matrix formed by deleting row r and column c fromis called the adjugate matrix of , written as adj[A].

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Finding inverse matrixLet A be an n×n matrix. Then

where Adj(A) denotes the adjugate matrix of A, det(A) is the determinant, and I is the identity matrix. If det(A) isinvertible in R, then the inverse matrix of A is

If R is a field (such as the field of real numbers), then this gives a formula for the inverse of A, provided det(A) ≠ 0.In fact, this formula will work whenever R is a commutative ring, provided that det(A) is a unit. If det(A) is not aunit, then A is not invertible.

Applications

Explicit formulas for small systems

Consider the linear system which in matrix format is Then, x and y can

be found with Cramer's rule as

and

The rules for 3×3 are similar. Given which in matrix format is

the values of x, y and z can be found as follows:

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Differential geometryCramer's rule is also extremely useful for solving problems in differential geometry. Consider the two equations

and . When u and v are independent variables, we can defineand

Finding an equation for is a trivial application of Cramer's rule.

First, calculate the first derivatives of F, G, x, and y:

Substituting dx, dy into dF and dG, we have:

Since u, v are both independent, the coefficients of du, dv must be zero. So we can write out equations for thecoefficients:

Now, by Cramer's rule, we see that:

This is now a formula in terms of two Jacobians:

Similar formulae can be derived for , ,

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AlgebraCramer's rule can be used to prove the Cayley–Hamilton theorem of linear algebra, as well as Nakayama's lemma,which is fundamental in commutative ring theory.

Integer programmingCramer's rule can be used to prove that an integer programming problem whose constraint matrix is totallyunimodular and whose right-hand side is integer, has integer basic solutions. This makes the integer programsubstantially easier to solve.

Ordinary differential equationsCramer's rule is used to derive the general solution to an inhomogeneous linear differential equation by the methodof variation of parameters.

Geometric interpretation

Geometric interpretation of Cramer's rule. The areas of the second andthird shaded parallelograms are the same and the second is times

the first. From this equality Cramer's rule follows.

Cramer's rule has a geometric interpretation that canbe considered also a proof or simply giving insightabout its geometric nature. These geometricarguments work in general and not only in the caseof two equations with two unknowns presented here.Given the system of equations

it can be considered as an equation between vectors

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The area of the parallelogram determined by and is given by the determinant of the system of

equations:

In general, when there are more variables and equations, the determinant of vectors of length will give thevolume of the parallelepiped determined by those vectors in the -th dimensional Euclidean space.

Therefore the area of the parallelogram determined by and has to be times the area of the

first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's

principle, has the same area as the parallelogram determined by and .

Equating the areas of this last and the second parallelogram gives the equation

from which Cramer's rule follows.

Incompatible and indeterminate casesA system of equations is said to be incompatible when there are no solutions and it is called indeterminate whenthere is more than one solution. For linear equations, an indeterminate system will have infinitely many solutions (ifit is over an infinite field), since the solutions can be expressed in terms of one or more parameters that can takearbitrary values.Cramer's rule applies to the case where the coefficient determinant is nonzero. In the contrary case the system iseither incompatible or indeterminate, based on the values of the determinants only for 2x2 systems.For 3x3 or higher systems, the only thing one can say when the coefficient determinant equals zero is: if any of the"numerator" determinants are nonzero, then the system must be incompatible. However, the converse is false: havingall determinants zero does not imply that the system is indeterminate. A simple example where all determinantsvanish but the system is still incompatible is the 3x3 system x+y+z=1, x+y+z=2, x+y+z=3.

Notes[1] Carl B. Boyer, A History of Mathematics, 2nd edition (Wiley, 1968), p. 431.[2] Victor Katz, A History of Mathematics, Brief edition (Pearson Education, 2004), pp. 378–379.[3] Ken Habgood, Itamar Arel, A condensation-based application of Cramerʼs rule for solving large-scale linear systems, Journal of Discrete

Algorithms, ISSN 1570-8667, 10.1016/j.jda.2011.06.007.

External links• Proof of Cramer's Rule (http:/ / planetmath. org/ encyclopedia/ ProofOfCramersRule. html)• WebApp descriptively solving systems of linear equations with Cramer's Rule (http:/ / sole. ooz. ie/ )• Cramer's Rule Solver (http:/ / www. cramersrule. info/ index. html)• Online Calculator of System of linear equations (http:/ / www. stud. feec. vutbr. cz/ ~xvapen02/ vypocty/ linrov.

php?language=english)

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Crystallographic restriction theorem 76

Crystallographic restriction theoremThe crystallographic restriction theorem in its basic form was based on the observation that the rotationalsymmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occurwith other symmetries, such as 5-fold; these were not discovered until 1984[1]

Prior to the discovery of quasicrystals, crystals were modeled as discrete lattices, generated by a list of independentfinite translations (Coxeter 1989). Because discreteness requires that the spacings between lattice points have a lowerbound, the group of rotational symmetries of the lattice at any point must be a finite group. The strength of thetheorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only afinite number of compatible groups.

Dimensions 2 and 3The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and we cantreat them together.

Lattice proofA rotation symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the sameplane, generating a regular polygon of coplanar lattice points. We now confine our attention to the plane in which thesymmetry acts (Scherrer 1946). (We might call this a proof in the style of Busby Berkeley, with lattice vectors ratherthan pretty ladies dancing and swirling in geometric patterns.)

Lattices restrict polygonsCompatible: 6-fold (3-fold), 4-fold (2-fold)

Incompatible: 8-fold, 5-fold

Now consider an 8-fold rotation, and thedisplacement vectors between adjacent points ofthe polygon. If a displacement exists betweenany two lattice points, then that samedisplacement is repeated everywhere in thelattice. So collect all the edge displacements tobegin at a single lattice point. The edge vectorsbecome radial vectors, and their 8-foldsymmetry implies a regular octagon of latticepoints around the collection point. But this isimpossible, because the new octagon is about80% smaller than the original. The significanceof the shrinking is that it is unlimited. The sameconstruction can be repeated with the newoctagon, and again and again until the distancebetween lattice points is as small as we like; thusno discrete lattice can have 8-fold symmetry.The same argument applies to any k-foldrotation, for k greater than 6.

A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, thenwe can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning tothe starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, butabout 60% smaller than the original.

Thus the theorem is proved.

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Crystallographic restriction theorem 77

The existence of quasicrystals and Penrose tilings shows that the assumption of a linear translation is necessary.Penrose tilings may have 5-fold rotational symmetry and a discrete lattice, and any local neighborhood of the tiling isrepeated infinitely many times, but there is no linear translation for the tiling as a whole. And without the discretelattice assumption, the above construction not only fails to reach a contradiction, but produces a (non-discrete)counterexample. Thus 5-fold rotational symmetry cannot be eliminated by an argument missing either of thoseassumptions. A Penrose tiling of the whole (infinite) plane can only have exact 5-fold rotational symmetry (of thewhole tiling) about a single point, however, whereas the 4-fold and 6-fold lattices have infinitely many centres ofrotational symmetry.

Trigonometry proofConsider two lattice points A and B separated by a translation vector r. Consider an angle α such that a rotation ofangle α about any lattice point is a symmetry of the lattice. Rotating about point B by α maps point A to a new pointA'. Similarly, rotating about point A by α, in the opposite direction, maps B to a point B'. Since both rotationsmentioned are symmetry operations, A' and B' must both be lattice points. Due to periodicity of the crystal, the newvector r' which connects them must be equal to an integral multiple of r:r = mr'The four translation vectors (three of which are given by r, and one which connects A' and B' given by r') form aparallelogram. Therefore, the length of r' is also given by:r = -2rcos(α) + r'Combining the two equations gives:cos(α) = (1 - m)/2 = M/2

Where M = 1-m is also an integer. Since we always have:| cos(α) | ≤ 1it follows that:| M | ≤ 2It can be shown that the only values of α in the 0° to 180° range that satisfy the three equations above are 0°, 60°,90°, 120°, and 180°. In terms of radians, the only allowed rotations consistent with lattice periodicity are given by2π/n, where n = 1, 2, 3, 4, 6. This corresponds to 1-, 2-, 3-, 4-, and 6-fold symmetry, respectively, and thereforeexcludes the possibility of 5-fold or greater than 6-fold symmetry.

Short trigonometry proofConsider a line of atoms A-O-B, separated by distance t. Rotate the entire row by +2π/n and -2π/n, with point O keptfixed. Due to the assumed periodicity of the lattice, the two lattice points C and D will be also in a directly below theinitial row; moreover C and D will be separated by r = mt, with m an integer. But by the geometry, the separationbetween these points is 2tcos(2π/n). Equating the two relations gives 2cos(2π/n) = m. This is satisfied by only n = 1,2, 3, 4, 6. (See also the sketch in section Preuve mathématique simple of the French Wikipedia article (Théorème derestriction cristallographique)

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Crystallographic restriction theorem 78

Matrix proofFor an alternative proof, consider matrix properties. The sum of the diagonal elements of a matrix is called the traceof the matrix. In 2D and 3D every rotation is a planar rotation, and the trace is a function of the angle alone. For a 2Drotation, the trace is 2 cos θ; for a 3D rotation, 1 + 2 cos θ.Examples

• Consider a 60° (6-fold) rotation matrix with respect to an orthonormal basis in 2D.

The trace is precisely 1, an integer.• Consider a 45° (8-fold) rotation matrix.

The trace is 2/√2, not an integer.Using a lattice basis, neither orthogonality nor unit length is guaranteed, only independence. However, the trace isthe same with respect to any basis. (Similarity transforms preserve trace.) In a lattice basis, because the rotation mustmap lattice points to lattice points, each matrix entry — and hence the trace — must be an integer. Thus, forexample, wallpaper and crystals cannot have 8-fold rotational symmetry. The only possibilities are multiples of 60°,90°, 120°, and 180°, corresponding to 6-, 4-, 3-, and 2-fold rotations.Example

• Consider a 60° (360°/6) rotation matrix with respect to the oblique lattice basis for a tiling by equilateral triangles.

The trace is still 1. The determinant (always +1 for a rotation) is also preserved.The general crystallographic restriction on rotations does not guarantee that a rotation will be compatible with aspecific lattice. For example, a 60° rotation will not work with a square lattice; nor will a 90° rotation work with arectangular lattice.

Higher dimensionsWhen the dimension of the lattice rises to four or more, rotations need no longer be planar; the 2D proof isinadequate. However, restrictions still apply, though more symmetries are permissible. For example, the hypercubiclattice has an eightfold rotational symmetry, corresponding to an eightfold rotational symmetry of the hypercube.This is of interest, not just for mathematics, but for the physics of quasicrystals under the cut-and-project theory. Inthis view, a 3D quasicrystal with 8-fold rotation symmetry might be described as the projection of a slab cut from a4D lattice.The following 4D rotation matrix is the aforementioned eightfold symmetry of the hypercube (and thecross-polytope):

Transforming this matrix to the new coordinates given by

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Crystallographic restriction theorem 79

will produce:

This third matrix then corresponds to a rotation both by 45° (in the first two dimensions) and by 135° (in the lasttwo). Projecting a slab of hypercubes along the first two dimensions of the new coordinates produces anAmmann–Beenker tiling (another such tiling is produced by projecting along the last two dimensions), whichtherefore also has 8-fold rotational symmetry on average.The A4 lattice and F4 lattice have order 10 and order 12 rotational symmetries, respectively.To state the restriction for all dimensions, it is convenient to shift attention away from rotations alone andconcentrate on the integer matrices (Bamberg, Cairns & Kilminster 2003). We say that a matrix A has order k whenits k-th power (but no lower), Ak, equals the identity. Thus a 6-fold rotation matrix in the equilateral triangle basis isan integer matrix with order 6. Let OrdN denote the set of integers that can be the order of an N×N integer matrix. Forexample, Ord2 = {1, 2, 3, 4, 6}. We wish to state an explicit formula for OrdN.Define a function ψ based on Euler's totient function φ; it will map positive integers to non-negative integers. For anodd prime, p, and a positive integer, k, set ψ(pk) equal to the totient function value, φ(pk), which in this case ispk−pk−1. Do the same for ψ(2k) when k > 1. Set ψ(2) and ψ(1) to 0. Using the fundamental theorem of arithmetic, wecan write any other positive integer uniquely as a product of prime powers, m = ∏α pα

k α; set ψ(m) = ∑α ψ(pαk α).

This differs from the totient itself, because it is a sum instead of a product.The crystallographic restriction in general form states that OrdN consists of those positive integers m such that ψ(m)≤ N.

Smallest dimension for a given order A080737

m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ψ(m) 0 0 2 2 4 2 6 4 6 4 10 4 12 6 6 8 16 6 18 6 8 10 22 6 20 12 18 8 28 6 30

Note that these additional symmetries do not allow a planar slice to have, say, 8-fold rotation symmetry. In the plane,the 2D restrictions still apply. Thus the cuts used to model quasicrystals necessarily have thickness.Integer matrices are not limited to rotations; for example, a reflection is also a symmetry of order 2. But by insistingon determinant +1, we can restrict the matrices to proper rotations.

Formulation in terms of isometriesThe crystallographic restriction theorem can be formulated in terms of isometries of Euclidean space. A set ofisometries can form a group. By a discrete isometry group we will mean an isometry group that maps every point toa discrete subset of RN, i.e. a set of isolated points. With this terminology, the crystallographic restriction theorem intwo and three dimensions can be formulated as follows.

For every discrete isometry group in two- and three-dimensional space which includes translations spanningthe whole space, all isometries of finite order are of order 1, 2, 3, 4 or 6.

Note that isometries of order n include, but are not restricted to, n-fold rotations. The theorem also excludes S8, S12,D4d, and D6d (see point groups in three dimensions), even though they have 4- and 6-fold rotational symmetry only.

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Note also that rotational symmetry of any order about an axis is compatible with translational symmetry along thataxis.The result in the table above implies that for every discrete isometry group in four- and five-dimensional spacewhich includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4, 5, 6, 8, 10,or 12.All isometries of finite order in six- and seven-dimensional space are of order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15,18, 20, 24 or 30 .

Notes[1] Shechtman et al (1984)

References• Bamberg, John; Cairns, Grant; Kilminster, Devin (March 2003), "The crystallographic restriction, permutations,

and Goldbach's conjecture" (http:/ / www. latrobe. edu. au/ mathstats/ staff/ cairns/ papers/ 42. pdf), AmericanMathematical Monthly 110 (3): 202–209, doi:10.2307/3647934, JSTOR 3647934

• Elliott, Stephen (1998), The Physics and Chemistry of Solids, Wiley, ISBN 0-471-98194-X• Coxeter, H. S. M. (1989), Introduction to Geometry (2nd ed.), Wiley, ISBN 978-0-471-50458-0• Scherrer, W. (1946), "Die Einlagerung eines regulären Vielecks in ein Gitter" (http:/ / www-gdz. sub.

uni-goettingen. de/ cgi-bin/ digbib. cgi?PPN378850199_0001), Elemente der Mathematik 1 (6): 97–98• Shechtman, D.; Blech, I.; Gratias, D.; Cahn, JW (1984), "Metallic phase with long-range orientational order and

no translational symmetry", Physical Review Letters 53 (20): 1951–1953, Bibcode 1984PhRvL..53.1951S,doi:10.1103/PhysRevLett.53.1951

External links• The crystallographic restriction (http:/ / www-history. mcs. st-and. ac. uk/ ~john/ geometry/ Lectures/ A2. html)

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Descartes' rule of signs 81

Descartes' rule of signsIn mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a techniquefor determining the number of positive or negative real roots of a polynomial.The rule gives us an upper bound number of positive or negative roots of a polynomial. It is not a deterministic rule,i.e. it does not tell the exact number of positive or negative roots.

Descartes' rule of signs

Positive rootsThe rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descendingvariable exponent, then the number of positive roots of the polynomial is either equal to the number of signdifferences between consecutive nonzero coefficients, or is less than it by a multiple of 2. Multiple roots of the samevalue are counted separately.

Negative rootsAs a corollary of the rule, the number of negative roots is the number of negative integers after negating thecoefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), orfewer than it by a multiple of 2.

ExampleFor example, the polynomial

has one sign change between the second and third terms (++, +−, −−). Therefore it has exactly one positive root.Note that the leading sign needs to be considered although it doesn't affect the answer in this case. In fact, thispolynomial factors as

so the roots are −1 (twice) and 1.Now consider the polynomial

This polynomial has two sign changes (−+, ++, +−), meaning the original polynomial has two or zero negative rootsand this second polynomial has two or zero positive roots. The factorization of the second polynomial is

So here, the roots are 1 (twice) and −1, the negation of the roots of the original polynomial. Since any nth degreepolynomial equation has exactly n roots, the minimum number of complex roots is equal to

Where p denotes the maximum number of positive roots, and q denotes the maximum number of negative roots (bothof which can be found out using descarte's rule of sign), and n denotes the degree of the equation.

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Descartes' rule of signs 82

Special caseThe exclusion of multiples of 2 is because the polynomial may have complex roots which always come in pairs.Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive andnegative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determinedin this case.

GeneralizationsIf the real polynomial P has k real positive roots counted with multiplicity, then for every a > 0 there are at least kchanges of sign in the sequence of coefficients of the Taylor series of the function eaxP(x).[1]

In the 1970s Askold Georgevich Khovanskiǐ developed the theory of fewnomials that generalises Descartes' rule.[2]

The rule of signs can be thought of as stating that the number of real roots of a polynomial is dependent on thepolynomial's complexity, and that this complexity is proportional to the number of monomials it has, not its degree.Khovanskiǐ showed that this holds true not just for polynomials but for algebraic combinations of manytranscendental functions, the so-called Pfaffian functions.

Notes[1] Vladimir P. Kostov, A mapping defined by the Schur-Szegő composition, Comptes Rendus Acad. Bulg. Sci. tome 63, No. 7, 2010, 943 - 952.[2] A. G. Khovanskii, Fewnomials, Princeton University Press (1991) ISBN 0821845470.

External links• Descartes’ Rule of Signs (http:/ / www. cut-the-knot. org/ fta/ ROS2. shtml) — Proof of the RuleThis article incorporates material from Descartes' rule of signs on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Page 88: Theorems in Algebra

Dirichlet's unit theorem 83

Dirichlet's unit theoremIn mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Gustav LejeuneDirichlet.[1] It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. Theregulator is a positive real number that determines how "dense" the units are.

Dirichlet's unit theoremThe statement is that the group of units is finitely generated and has rank (maximal number of multiplicativelyindependent elements) equal to

r = r1 + r2 − 1where r1 is the number of real embeddings and r2 the number of conjugate pairs of complex embeddings of K. Thischaracterisation of r1 and r2 is based on the idea that there will be as many ways to embed K in the complex numberfield as the degree n = [K : Q]; these will either be into the real numbers, or pairs of embeddings related by complexconjugation, so that

n = r1 + 2r2.Other ways of determining r1 and r2 are• use the primitive element theorem to write K = Q(α), and then r1 is the number of conjugates of α that are real,

2r2 the number that are complex;• write the tensor product of fields K ⊗

QR as a product of fields, there being r1 copies of R and r2 copies of C.

As an example, if K is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadraticfield. The theory for real quadratic fields is essentially the theory of Pell's equation.The rank is > 0 for all number fields besides Q and imaginary quadratic fields, which have rank 0. The 'size' of theunits is measured in general by a determinant called the regulator. In principle a basis for the units can be effectivelycomputed; in practice the calculations are quite involved when n is large.The torsion in the group of units is the set of all roots of unity of K, which form a finite cyclic group. For a numberfield with at least one real embedding the torsion must therefore be only {1,−1}. There are number fields, forexample most imaginary quadratic fields, having no real embeddings which also have {1,−1} for the torsion of itsunit group.Totally real fields are special with respect to units. If L/K is a finite extension of number fields with degree greaterthan 1 and the units groups for the integers of L and K have the same rank then K is totally real and L is a totallycomplex quadratic extension. The converse holds too. (An example is K equal to the rationals and L equal to animaginary quadratic field; both have unit rank 0.)There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structureof the group of S-units, determining the rank of the unit group in localizations of rings of integers. Also, the Galoismodule structure of has been determined.[2]

The regulatorSuppose that u1,...,ur are a set of generators for the unit group modulo roots of unity. If u is an algebraic number, write u1, ..., ur+1 for the different embeddings into R or C, and write Ni for the degree of the corresponding embedding over R (so it is 1 for real embeddings and 2 for complex ones). Then the r by r + 1 matrix whose entries are has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries of a row). This implies that the absolute value R of the determinant of the submatrix formed by deleting one column is independent of the column. The number R is called the regulator of the algebraic

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Dirichlet's unit theorem 84

number field (it does not depend on the choice of generators ui). It measures the "density" of the units: if theregulator is small, this means that there are "lots" of units.The regulator has the following geometric interpretation. The map taking a unit u to the vector with entries Nilog|ui|has image in the r-dimensional subspace of Rr+1 consisting of all vectors whose entries have sum 0, and byDirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice isR√(r+1).The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, thoughthere are now computer algebra packages that can do it in many cases. It is usually much easier to calculate theproduct hR of the class number h and the regulator using the class number formula, and the main difficulty incalculating the class number of an algebraic number field is usually the calculation of the regulator.

Examples

A fundamental domain in logarithmic space of the group of units of the cycliccubic field K obtained by adjoining to Q a root of f(x) = x3 + x2 − 2x − 1. If α

denotes a root of f(x), then a set of fundamental units is {ε1, ε2} whereε1 = α2 + α − 1 and ε2 = 2 − α2. The area of the fundamental domain is

approximately 0.910114, so the regulator of K is approximately 0.525455.

• The regulator of an imaginary quadraticfield, or of the rational integers, is 1 (asthe determinant of a 0×0 matrix is 1).

• The regulator of the real quadratic fieldQ(√5) is log((√5 + 1)/2). This can beseen as follows. A fundamental unit is(√5 + 1)/2, and its images under the twoembeddings into R are (√5 + 1)/2 and(−√5 + 1)/2. So the r by r + 1 matrix is

• The regulator of the cyclic cubic field Q(α), where α is a root of x3 + x2 − 2x − 1, is approximately 0.5255. Abasis of the group of units modulo roots of unity is {ε1, ε2} where ε1 = α2 + α − 1 and ε2 = 2 − α2.[3]

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Dirichlet's unit theorem 85

Higher regulatorsA 'higher' regulator refers to a construction for an algebraic K-group with index n > 1 that plays the same role as theclassical regulator does for the group of units, which is a group K1. A theory of such regulators has been indevelopment, with work of Armand Borel and others. Such higher regulators play a role, for example, in theBeilinson conjectures, and are expected to occur in evaluations of certain L-functions at integer values of theargument.

Stark regulatorThe formulation of Stark's conjectures led Harold Stark to define what is now called the Stark regulator, similar tothe classical regulator as a determinant of logarithms of units, attached to any Artin representation.[4] [5]

Notes[1] Elstrodt 2007, §8.D[2] Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.[3] Cohen 1993, Table B.4[4] PDF (http:/ / www. math. tifr. res. in/ ~dprasad/ artin. pdf)[5] PDF (http:/ / www. math. harvard. edu/ ~dasgupta/ papers/ Dasguptaseniorthesis. pdf)

References• Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics.

138. Berlin, New York: Springer-Verlag. ISBN 978-3-540-55640-4. MR1228206• Elstrodt, Jürgen (2007). "The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (http:/ / www. uni-math.

gwdg. de/ tschinkel/ gauss-dirichlet/ elstrodt-new. pdf) (PDF). Clay Mathematics Proceedings. Retrieved2010-06-13.

• Serge Lang, Algebraic number theory, ISBN 0-387-94225-4• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322,

Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR1697859• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der

Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196

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Engel theorem 86

Engel theoremIn representation theory, Engel's theorem is one of the basic theorems in the theory of Lie algebras; it asserts thatfor a Lie algebra two concepts of nilpotency are identical. A useful form of the theorem says that if a Lie algebra Lof matrices consists of nilpotent matrices, then they can all be simultaneously brought to a strictly upper triangularform. The theorem is named after the mathematician Friedrich Engel, who sketched a proof of it in a letter to Killingdated 20 July 1890 (Hawkins 2000, p. 176). Engel's student Umlauf gave a complete proof in his 1891 dissertation,reprinted as (Umlauf 2010).A linear operator T on a vector space V is defined to be nilpotent if there is a positive integer k such that Tk = 0. Forexample, any operator given by a matrix whose entries are zero on and below its diagonal is nilpotent.

An element x of a Lie algebra L is ad-nilpotent if and only if the linear operator on L defined by

is nilpotent. Note that in the Lie algebra L(V) of linear operators on V, the identity operator IV is ad-nilpotent(because ) but is not a nilpotent operator.A Lie algebra L is nilpotent if and only if the lower central series defined recursively by

eventually reaches {0}.Theorem. A finite-dimensional Lie algebra L is nilpotent if and only if every element of L is ad-nilpotent.Note that no assumption on the underlying base field is required.The key lemma in the proof of Engel's theorem is the following fact about Lie algebras of linear operators on finitedimensional vector spaces which is useful in its own right:Let L be a Lie subalgebra of L(V). Then L consists of nilpotent operators if and only if there is a sequence

of subspaces of V such that , and

Thus Lie algebras of nilpotent operators are simultaneously strictly upper-triangulizable.

References• Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0• Hawkins, Thomas (2000), Emergence of the theory of Lie groups [1], Sources and Studies in the History of

Mathematics and Physical Sciences, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98963-1, MR1771134• G. Hochschild, The Structure of Lie Groups, Holden Day, 1965.• J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972.• Umlauf, Karl Arthur (2010) [1891] (in German), Über Die Zusammensetzung Der Endlichen Continuierlichen

Transformationsgruppen, Insbesondre Der Gruppen Vom Range Null [2], Inaugural-Dissertation, Leipzig, NabuPress, ISBN 978-1-141-58889-3

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References[1] http:/ / books. google. com/ books?isbn=978-0-387-98963-1[2] http:/ / books. google. com/ books?isbn=978-1141588893

Factor theoremIn algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of thepolynomial remainder theorem.

The factor theorem states that a polynomial has a factor if and only if .

Factorization of polynomialsTwo problems where the factor theorem is commonly applied are those of factoring a polynomial and finding theroots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentiallyequivalent.The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact,thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:

1. "Guess" a zero of the polynomial . (In general, this can be very hard, but math textbook problems thatinvolve solving a polynomial equation are often designed so that some roots are easy to discover.)

2. Use the factor theorem to conclude that is a factor of .3. Compute the polynomial , for example using polynomial long division.4. Conclude that any root of is a root of . Since the polynomial degree of is one

less than that of , it is "simpler" to find the remaining zeros by studying .

ExampleYou wish to find the factors at

To do this you would use trial and error to find the first x value that causes the expression to equal zero. To find outif is a factor, substitute into the polynomial above:

As this is equal to 18 and not 0 this means is not a factor of . So, we next try(substituting into the polynomial):

This is equal to . Therefore , which is to say , is a factor, and is a root of

The next two roots can be found by algebraically dividing by to get a quadratic,which can be solved directly, by the factor theorem or by the quadratic equation.

and therefore and are the factors of

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Factor theorem 88

Formal versionLet be a polynomial with complex coefficients, and be in an integral domain (e.g. ). Then if and only if can be written in the form where is also a polynomial. isdetermined uniquely.This indicates that those for which are precisely the roots of . Repeated roots can be found byapplication of the theorem to the quotient , which may be found by polynomial long division.

Feit–Thompson theoremIn mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order issolvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963)

HistoryThe contrast that these results shew between groups of odd and even order suggests inevitably that simple groups of odd order do notexist.

William Burnside (1911, p. 503 note M)

William Burnside (1911, p. 503 note M) conjectured that every nonabelian finite simple group has even order.Richard Brauer (1957) suggested using the centralizers of involutions of simple groups as the basis for theclassification of finite simple groups, as the Brauer-Fowler theorem shows that there are only a finite number offinite simple groups with given centralizer of an involution. A group of odd order has no involutions, so to carry outBrauer's program it is first necessary to show that non-cyclic finite simple groups never have odd order. This isequivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groupssuch that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groupsof odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups suchthat the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groupsof Lie type in the process, that are now called Suzuki groups.)Feit, Hall, and Thompson (1960) extended Suzuki's work to the family of CN groups; these are groups such that theCentralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable.Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long fora proof in group theory.The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclicsimple group of odd order such that every proper subgroup is solvable. This proves that every finite group of oddorder is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable.Although the proof follows the same general outline as the CA theorem and the CN theorem, the details are vastlymore complicated. The final paper is 255 pages long.

Significance of the proofThe Feit–Thompson theorem showed that the classification of finite simple groups using centralizers of involutions might be possible, as every nonabelian simple group has an involution. Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification. Perhaps the most revolutionary aspect of the proof was its length: before the Feit-Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day. Once group theorists realized that such long arguments could work, a series of papers that were several hundred pages long started to appear. Some of these

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dwarfed even the Feit–Thompson paper; Aschbacher and Smith's paper on quasithin groups was 1221 pages long.

Revision of the proofMany mathematicians have simplified parts of the original Feit–Thompson proof. However all of theseimprovements are in some sense local; the global structure of the argument is still the same, but some of the detailsof the arguments have been simplified.The simplified proof has been published in two books: (Bender & Glauberman 1995), which covers everythingexcept the character theory, and (Peterfalvi 2000, part I) which covers the character theory. This revised proof is stillvery hard, and is longer than the original proof, but is written in a more leisurely style.

An outline of the proofInstead of describing the Feit–Thompson theorem directly, it is easier to describe Suzuki's CA theorem and thencomment on some of the extensions needed for the CN-theorem and the odd order theorem. The proof can be brokenup into three steps. We let G be a non-abelian (minimal) simple group of odd order satisfying the CA condition. Fora more detailed exposition of the odd order paper see Thompson (1963) or (Gorenstein 1980) or Glauberman (1999).Step 1. Local analysis of the structure of the group G. This is easy in the CA case because the relation "a commuteswith b" is an equivalence relation on the non-identity elements. So the elements break up into equivalence classes,such that each equivalence class is the set of non-identity elements of a maximal abelian subgroup. The normalizersof these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of G. These normalizers areFrobenius groups whose character theory is reasonably transparent, and well-suited to manipulations involvingcharacter induction. Also, the set of prime divisors of |G| is partitioned according to the primes which divide theorders of the distinct conjugacy classes of maximal abelian subgroups of |G|. This pattern of partitioning the primedivisors of |G| according to conjugacy classes of nilpotent Hall subgroups (a Hall subgroup is one whose order andindex are relatively prime) whose normalizers give all the maximal subgroups of G (up to conjugacy) is repeated inboth the proof of the Feit-Hall-Thompson CN-theorem and in the proof of the Feit-Thompson odd-order theorem.The proof of the CN-case is already considerably more difficult than the CA-case, while this part of the proof of theodd-order theorem takes over 100 journal pages. (Bender later simplified this part of the proof using Bender'smethod.) Whereas in the CN-case, the resulting maximal subgroups are still Frobenius groups, the maximalsubgroups which occur in the proof of the odd-order theorem need no longer have this structure, and the analysis oftheir structure and interplay produces 5 very complicated possible configurations. Peterfalvi (2000) used the Dadeisometry to simplify the character theory.Step 2. Character theory of G. If X is an irreducible character of the normalizer H of the maximal abelian subgroup Aof the CA group G, not containing A in its kernel, we can induce X to a character Y of G, which is not necessarilyirreducible. Because of the known structure of G, it is easy to find the character values of Y on all but the identityelement of G. This implies that if X1 and X2 are two such irreducible characters of H and Y1 and Y2 are thecorresponding induced characters, then Y1 − Y2 is completely determined, and calculating its norm shows that it isthe difference of two irreducible characters of G (these are sometimes known as exceptional characters of G withrespect to H). A counting argument shows that each non-trivial irreducible character of G arises exactly once as anexceptional character associated to the normalizer of some maximal abelian subgroup of G. A similar argument (butreplacing abelian Hall subgroups by nilpotent Hall subgroups) works in the proof of the CN-theorem. However, inthe proof of the odd-order theorem, the arguments for constructing characters of G from characters of subgroups arefar more delicate, and involve more subtle maps between character rings than character induction, since the maximalsubgroups have a more complicated structure and are embedded in a less transparent way.Step 3. By step 2, we have a complete and precise description of the character table of the CA group G. From this, and using the fact that G has odd order, sufficient information is available to obtain estimates for |G| and arrive at a

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contradiction to the assumption that G is simple. This part of the argument works similarly in the CN-group case.In the proof of the Feit–Thompson theorem, however, this step is (as usual) vastly more complicated. The charactertheory only eliminates four of the possible five configurations left after step 1. To eliminate the final case, Thompsonused some fearsomely complicated manipulations with generators and relations (which were later simplified byPeterfalvi (1984), whose argument is reproduced in (Bender & Glauberman 1994). The Feit-Thompson conjecturewould simplify this step if it were proven.

References• Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical

Society Lecture Note Series, 188, Cambridge University Press, ISBN 978-0-521-45716-3, MR1311244• Brauer, R. (1957), "On the structure of groups of finite order" [1], Proceedings of the International Congress of

Mathematicians, Amsterdam, 1954, Vol. 1, Erven P. Noordhoff N.V., Groningen, pp. 209–217, MR0095203• Burnside, William (1911), Theory of groups of finite order [2], New York: Dover Publications,

ISBN 978-0-486-49575-0, MR0069818• Feit, Walter; Thompson, John G.; Hall, Marshall, Jr. (1960), "Finite groups in which the centralizer of any

non-identity element is nilpotent", Math. Z. 74: 1–17, doi:10.1007/BF01180468, MR0114856• Feit, Walter; Thompson, John G. (1962), "A solvability criterion for finite groups and some consequences", Proc.

Nat. Acad. Sci. 48 (6): 968–970, doi:10.1073/pnas.48.6.968, JSTOR 71265, MR0143802• Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order" [3], Pacific Journal of Mathematics

13: 775–1029, ISSN 0030-8730, MR0166261• Glauberman, George (1999), "A new look at the Feit-Thompson odd order theorem" [4], Matemática

Contemporânea 16: 73–92, ISSN 0103-9059, MR1756828• Gorenstein, D. (1980), Finite groups [1] (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6,

MR569209• Peterfalvi, Thomas (1984), "Simplification du chapitre VI de l'article de Feit et Thompson sur les groupes d'ordre

impair", Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique 299 (12): 531–534,ISSN 0249-6291, MR770439

• Peterfalvi, Thomas (2000), Character theory for the odd order theorem [5], London Mathematical Society LectureNote Series, 272, Cambridge University Press, ISBN 978-0-521-64660-4, MR1747393

• Suzuki, Michio (1957), "The nonexistence of a certain type of simple groups of odd order", Proceedings of theAmerican Mathematical Society (Proceedings of the American Mathematical Society, Vol. 8, No. 4) 8 (4):686–695, doi:10.2307/2033280, JSTOR 2033280, MR0086818

• Thompson, John G. (1963), "Two results about finite groups" [6], Proc. Internat. Congr. Mathematicians(Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 296–300, MR0175972

References[1] http:/ / mathunion. org/ ICM/ ICM1954. 1/[2] http:/ / books. google. com/ books?isbn=1440035458[3] http:/ / projecteuclid. org/ Dienst/ UI/ 1. 0/ Journal?authority=euclid. pjm& issue=1103053941[4] http:/ / www. mat. unb. br/ ~matcont/ 16_5. ps[5] http:/ / books. google. com/ books?isbn=052164660X[6] http:/ / mathunion. org/ ICM/ ICM1962. 1/

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Fitting's theorem 91

Fitting's theoremFitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows:

If M and N are nilpotent normal subgroups of a group G, then their product MN is also a nilpotent normalsubgroup of G; if, moreover, M is nilpotent of class m and N is nilpotent of class n, then MN is nilpotent ofclass at most m + n.

By induction it follows also that the subgroup generated by a finite collection of nilpotent normal subgroups isnilpotent. This can be used to show that the Fitting subgroup of certain types of groups (including all finite groups) isnilpotent. However, a subgroup generated by an infinite collection of nilpotent normal subgroups need not benilpotent.

Order-theoretic statementIn terms of order theory, (part of) Fitting's theorem can be stated as:

The set of nilpotent normal subgroups form a lattice of subgroups.Thus the nilpotent normal subgroups of a finite group also form a bounded lattice, and have a top element, the Fittingsubgroup.However, nilpotent normal subgroups do not in general form a complete lattice, as a subgroup generated by aninfinite collection of nilpotent normal subgroups need not be nilpotent, though it will be normal. The join of allnilpotent normal subgroups is still defined as the Fitting subgroup, but it need not be nilpotent.

Focal subgroup theoremIn abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finitegroup. The focal subgroup theorem was introduced in (Higman 1958) and is the "first major application of thetransfer" according to (Gorenstein, Lyons & Solomon 1996, p. 90). The focal subgroup theorem relates the ideas oftransfer and fusion such as described in (Grün 1935). Various applications of these ideas include local criteria forp-nilpotence and various non-simplicity criterion focussing on showing that a finite group has a normal subgroup ofindex p.

BackgroundThe focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index apower of p, the transfer homomorphism, and fusion of elements.

SubgroupsThe following three normal subgroups of index a power of p are naturally defined, and arise as the smallest normalsubgroups such that the quotient is (a certain kind of) p-group. Formally, they are kernels of the reflection onto thereflective subcategory of p-groups (respectively, elementary abelian p-groups, abelian p-groups).• Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian group, and is the

largest elementary abelian p-group onto which G surjects.• Ap(G) (notation from (Isaacs 2008, 5D, p. 164)) is the intersection of all normal subgroups K such that G/K is an

abelian p-group (i.e., K is an index normal subgroup that contains the derived group ): G/Ap(G) is thelargest abelian p-group (not necessarily elementary) onto which G surjects.

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• Op(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., Kis an index normal subgroup): G/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects.Op(G) is also known as the p-residual subgroup.

Firstly, as these are weaker conditions on the groups K, one obtains the containmentsThese are further related as:

Ap(G) = Op(G)[G,G].Op(G) has the following alternative characterization as the subgroup generated by all Sylow q-subgroups of G asq≠p ranges over the prime divisors of the order of G distinct from p.Op(G) is used to define the lower p-series of G, similarly to the upper p-series described in p-core.

Transfer homomorphismThe transfer homomorphism is a homomorphism that can be defined from any group G to the abelian groupH/[H,H] defined by a subgroup H ≤ G of finite index, that is [G:H] < ∞. The transfer map from a finite group G intoits Sylow p-subgroup has a kernel that is easy to describe:

The kernel of the transfer homomorphism from a finite group G into its Sylow p-subgroup P has Ap(G) as itskernel, (Isaacs 2008, Theorem 5.20, p. 165).

In other words, the "obvious" homomorphism onto an abelian p-group is in fact the most general suchhomomorphism.

FusionThe fusion pattern of a subgroup H in G is the equivalence relation on the elements of H where two elements h, k ofH are fused if they are G-conjugate, that is, if there is some g in G such that h = kg. The normal structure of G has aneffect on the fusion pattern of its Sylow p-subgroups, and conversely the fusion pattern of its Sylow p-subgroups hasan effect on the normal structure of G, (Gorenstein, Lyons & Solomon 1996, p. 89).

Focal subgroupIf one defines, as in (Gorenstein 1980, p. 246), the focal subgroup of P in G as the intersection P∩[G,G] of theSylow p-subgroup P of the finite group G with the derived subgroup [G,G] of G, then the focal subgroup is clearlyimportant as it is a Sylow p-subgroup of the derived subgroup. However, more importantly, one gets the followingresult:

There exists a normal subgroup K of G with G/K an abelian p-group isomorphic to P/P∩[G,G] (here K denotesAp(G)), andif K is a normal subgroup of G with G/K an abelian p-group, then P∩[G,G] ≤ K, and G/K is a homomorphicimage of P/P∩[G,G], (Gorenstein 1980, Theorem 7.3.1, p. 90).

One can define, as in (Isaacs 2008, p. 165) the focal subgroup of H with respect to G as:FocG(H) = ⟨ x−1 y | x,y in H and x is G-conjugate to y ⟩.

This focal subgroup measures the extent to which elements of H fuse in G, while the previous definition measuredcertain abelian p-group homomorphic images of the group G. The content of the focal subgroup theorem is that thesetwo definitions of focal subgroup are compatible.

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Statement of the theoremThe focal subgroup of a finite group X with Sylow p-subgroup P is given by:

P∩[G,G] = P∩Ap(G) = P∩ker(v) = FocG(P) = ⟨ x−1 y | x,y in P and x is G-conjugate to y ⟩where v is the transfer homomorphism from G to P/[P,P], (Isaacs 2008, Theorem 5.21, p. 165).

History and generalizationsThis connection between transfer and fusion is credited to (Higman 1958),[1] where, in different language, the focalsubgroup theorem was proved along with various generalizations. The requirement that G/K be abelian was dropped,so that Higman also studied Op(G) and the nilpotent residual γ∞(G), as so called hyperfocal subgroups. Higmanalso did not restrict to a single prime p, but rather allowed π-groups for sets of primes π and used Philip Hall'stheorem of Hall subgroups in order to prove similar results about the transfer into Hall π-subgroups; taking π = {p} aHall π-subgroup is a Sylow p-subgroup, and the results of Higman are as presented above.Interest in the hyperfocal subgroups was renewed by work of (Puig 2000) in understanding the modularrepresentation theory of certain well behaved blocks. The hyperfocal subgroup of P in G can defined as P∩γ∞(G)that is, as a Sylow p-subgroup of the nilpotent residual of G. If P is a Sylow p-subgroup of the finite group G, thenone gets the standard focal subgroup theorem:

P∩γ∞(G) = P∩Op(G) = ⟨ x−1 y : x,y in P and y = xg for some g in G of order coprime to p ⟩and the local characterization:

P∩Op(G) = ⟨ x−1 y : x,y in Q ≤ P and y = xg for some g in NG(Q) of order coprime to p ⟩.This compares to the local characterization of the focal subgroup as:

P∩Ap(G) = ⟨ x−1 y : x,y in Q ≤ P and y = xg for some g in NG(Q) ⟩.Puig is interested in the generalization of this situation to fusion systems, a categorical model of the fusion pattern ofa Sylow p-subgroup with respect to a finite group that also models the fusion pattern of a defect group of a p-blockin modular representation theory. In fact fusion systems have found a number of surprising applications andinspirations in the area of algebraic topology known as equivariant homotopy theory. Some of the major algebraictheorems in this area only have topological proofs at the moment.

Other characterizationsVarious mathematicians have presented methods to calculate the focal subgroup from smaller groups. For instance,the influential work (Alperin 1967) develops the idea of a local control of fusion, and as an example applicationshows that:

P ∩ Ap(G) is generated by the commutator subgroups [Q, NG(Q)] where Q varies over a family C ofsubgroups of P

The choice of the family C can be made in many ways (C is what is called a "weak conjugation family" in (Alperin1967)), and several examples are given: one can take C to be all non-identity subgroups of P, or the smaller choice ofjust the intersections Q = P ∩ Pg for g in G in which NP(Q) and NPg(Q) are both Sylow p-subgroups of NG(Q). Thelatter choice is made in (Gorenstein 1980, Theorem 7.4.1, p. 251). The work of (Grün 1935) studied aspects of thetransfer and fusion as well, resulting in Grün's first theorem:

P ∩ Ap(G) is generated by P∩[N,N] and P ∩ [Q, Q] where N = NG(P) and Q ranges over the set of Sylowp-subgroups Q = Pg of G (Gorenstein 1980, Theorem 7.4.2, p. 252).

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ApplicationsThe textbook presentations in (Rose 1978, pp. 254–264), (Isaacs 2008, Chapter 5), (Hall 1959, Chapter 14), (Suzuki1986, §5.2, pp. 138–165), all contain various applications of the focal subgroup theorem relating fusion, transfer, anda certain kind of splitting called p-nilpotence.During the course of the Alperin–Brauer–Gorenstein theorem classifying finite simple groups with quasi-dihedralSylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups:the 2-nilpotent groups, the Q-type groups whose focal subgroup is a generalized quaternion group of index 2, theD-type groups whose focal subgroup a dihedral group of index 2, and the QD-type groups whose focal subgroup isthe entire quasi-dihedral group. In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classesof cyclic subgroups of order 4; the Q-type have 2 classes of involutions and one class of cyclic subgroup of order 4;the QD-type have one class each of involutions and cyclic subgroups of order 4. In other words, finite groups withquasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according totheir fusion patterns. The explicit lists of groups with each fusion pattern are contained in (Alperin, Brauer &Gorenstein 1970).

Notes[1] The focal subgroup theorem and/or the focal subgroup is due to (Higman 1958) according to (Gorenstein, Lyons & Solomon 1996, p. 90),

(Rose 1978, p. 255), (Suzuki 1986, p. 141); however, the focal subgroup theorem as stated there and here is quite a bit older and alreadyappears in textbook form in (Hall 1959, p. 215). There and in (Puig 2000) the ideas are credited to (Grün 1935); compare to (Grün 1935, Satz5) in the special case of p-normal groups, and the general result in Satz 9 which is in some sense a refinement of the focal subgroup theorem.

References• Alperin, J. L. (1967), "Sylow intersections and fusion", Journal of Algebra 6: 222–241,

doi:10.1016/0021-8693(67)90005-1, ISSN 0021-8693, MR0215913• Alperin, J. L.; Brauer, R.; Gorenstein, D. (1970), "Finite groups with quasi-dihedral and wreathed Sylow

2-subgroups.", Transactions of the American Mathematical Society (American Mathematical Society) 151:1–261, doi:10.2307/1995627, ISSN 0002-9947, MR0284499

• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR81b:20002• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1996), The classification of the finite simple groups. Number

2. Part I. Chapter G (https:/ / www. ams. org/ online_bks/ surv402), Mathematical Surveys and Monographs, 40,Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0390-5, MR1358135

• Grün, Otto (1935), "Beiträge zur Gruppentheorie. I." (http:/ / resolver. sub. uni-goettingen. de/purl?GDZPPN002173409) (in German), Journal für Reine und Angewandte Mathematik 174: 1–14,ISSN 0075-4102, Zbl 0012.34102

• Hall, Marshall, Jr. (1959), The theory of groups, New York: Macmillan, MR0103215• Higman, Donald G. (1953), "Focal series in finite groups" (http:/ / www. cms. math. ca/ cjm/ v5/ p477), Canadian

Journal of Mathematics 5: 477–497, ISSN 0008-414X, MR0058597• Puig, Lluis (2000), "The hyperfocal subalgebra of a block", Inventiones Mathematicae 141 (2): 365–397,

doi:10.1007/s002220000072, ISSN 0020-9910, MR1775217• Rose, John S. (1994) [1978], A Course in Group Theory, New York: Dover Publications,

ISBN 978-0-486-68194-8, MR0498810• Suzuki, Michio (1986), Group theory. II, Grundlehren der Mathematischen Wissenschaften [Fundamental

Principles of Mathematical Sciences], 248, Berlin, New York: Springer-Verlag, ISBN 978-0-387-10916-9,MR815926

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Frobenius determinant theorem 95

Frobenius determinant theoremIn mathematics, the Frobenius determinant theorem is a discovery made in 1896 by the mathematician RichardDedekind, who wrote a letter to F. G. Frobenius about it (reproduced in (Dedekind 1968), with an English translationin (Curtis 2003, p.51)).If one takes the multiplication table of a group G and replaces each entry g with the variable xg, and subsequentlytakes the determinant, then the determinant factors as a product of n irreducible polynomials, where n is the numberof conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved thissurprising fact, and this theorem became known as the Frobenius determinant theorem.

Formal statementLet a finite group have elements , and let be associated with each element of . Definethe matrix with entries . Then

where r is the number of conjugacy classes of G.

References• Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer [1],

History of Mathematics, Providence, R.I.: American Mathematical Society, doi:10.1090/S0273-0979-00-00867-3,ISBN 978-0-8218-2677-5, MR1715145 Review [2]

• Dedekind, Richard (1968) [1931], Fricke, Robert; Noether, Emmy; Ore, öystein, eds., Gesammelte mathematischeWerke. Bände I--III, New York: Chelsea Publishing Co., JFM 56.0024.05, MR0237282

• Etingof, Pavel. Lectures on Representation Theory [3].• Frobenius, Ferdinand Georg (1968), Serre, J.-P., ed., Gesammelte Abhandlungen. Bände I, II, III, Berlin, New

York: Springer-Verlag, ISBN 978-3-540-04120-7, MR0235974

References[1] http:/ / books. google. com/ books?isbn=0821826778[2] http:/ / www. ams. org/ journals/ bull/ 2000-37-03/ S0273-0979-00-00867-3/[3] http:/ / www-math. mit. edu/ ~etingof/ cltrunc. pdf

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Frobenius theorem (real division algebras) 96

Frobenius theorem (real division algebras)In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand GeorgFrobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers.According to the theorem, every such algebra is isomorphic to one of the following:• R (the real numbers)• C (the complex numbers)• H (the quaternions).These algebras have dimensions 1, 2, and 4, respectively. Of these three algebras, the real and complex numbers arecommutative, but the quaternions are not.This theorem is closely related to Hurwitz's theorem, which states that the only normed division algebras over thereal numbers are R, C, H, and the (non-associative) algebra O of octonions.

ProofThe main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem ofalgebra.We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D byleft-multiplication and we will identify d with that endomorphism. Therefore we can speak about the trace of d, thecharacteristic and minimal polynomials. Also, we identify the real multiples of 1 with R. When we write foran element a of D, we tacitly assume that a is contained in R. The key to the argument is the followingClaim: The set V of all elements a of D such that is a vector subspace of D of codimension 1.To see that, we pick an a ∈ D. Let m be the dimension of D as an R-vector space. Let be the characteristicpolynomial of a. By the fundamental theorem of algebra, we can write

for some real ti and (non-real) complex numbers zj. We have . The polynomialsare irreducible over R. By the Cayley–Hamilton theorem, p(a) = 0

and because D is a division algebra, it follows that either for some i or that , for some j. The first case implies that a ∈ R. In the second case, it follows that isthe minimal polynomial of a. Because p(x) has the same complex roots as the minimal polynomial and because it isreal it follows that

and m=2k. The coefficient of in is the trace of a (up to sign). Therefore we read from that equation: thetrace of a is zero if and only if , that is .Therefore V is the subset of all a with tr a = 0. In particular, it is a vector subspace (!). Moreover, V has codimension1 since it is the kernel of a (nonzero) linear form. Also note that D is the direct sum of R and V (as vector spaces).Therefore, V generates D as an algebra.

Define now for Because of the identity , it follows that is real and since if . Thus B is a positive definitesymmetric bilinear form, in other words, an inner product on V.Let W be a subspace of V which generated D as an algebra and which is minimal with respect to that property. Let

be an orthonormal basis of W. These elements satisfy the following relations:

If n = 0, then D is isomorphic to R.

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If n = 1, then D is generated by 1 and e1 subject to the relation . Hence it is isomorphic to C.If n = 2, it has been shown above that D is generated by subject to the relations

and . These are precisely the relations for H.If n > 2, the D cannot be a division algebra. Assume that n > 2. Put . It is easy to see that (this only works if n > 2). Therefore implies that u= ±1 (because D is stillassumed to be a division algebra). But if u= ±1, then and so generates D. Thiscontradicts the minimality of W.Remark: The fact that D is generated by subject to above relation can be interpreted as the statementthat D is the Clifford algebra of Rn. The last step shows that the only real Clifford algebras which are divisionalgebras are Cl0, Cl1 and Cl2.Remark: As a consequence, the only commutative division algebras are R and C. Also note that H is not aC-algebra. If it were, then the center of H has to contain C, but the center of H is R. Therefore, the only divisionalgebra over C is C itself.

Pontryagin variantIf D is a connected, locally compact division ring, then either D = R, or D = C, or D = H.

References• Ray E. Artz (2009) Scalar Algebras and Quaternions [1], Theorem 7.1 "Frobenius Classification", page 26.• Ferdinand Georg Frobenius (1878) "Über lineare Substitutionen und bilineare Formen [2]", Journal für die reine

und angewandte Mathematik 84:1–63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I,pp.343–405.

• Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp.30–2 ISBN 0-7923-2459-5 .• Leonard Dickson (1914) Linear Algebras, Cambridge University Press. See §11 "Algebra of real quaternions; its

unique place among algebras", pages 10 to 12.• R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366–8.• Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.

References[1] http:/ / www. math. cmu. edu/ ~wn0g/ noll/ qu1. pdf[2] http:/ / commons. wikimedia. org/ wiki/ File:%C3%9Cber_lineare_Substitutionen_und_bilineare_Formen. djvu

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Fundamental lemma (Langlands program)In the theory of automorphic forms, an area of mathematics, the term fundamental lemma refers to any of a groupof results conjectured by Robert Langlands in the course of developing the Langlands program. A considerable bodyof subsequent work was conditional on their validity. The fundamental lemma was proved by Gérard Laumon andNgô Bảo Châu in the case of unitary groups and then by Ngô for general reductive groups, building on a series ofimportant reductions made by Jean-Loup Waldspurger to the case of Lie algebras. Time magazine placed Ngô'sproof on the list of the "Top 10 scientific discoveries of 2009".[1] [2] [3] In 2010 Ngô was awarded the Fields medalfor this proof.

Motivation and historyRobert Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur–Selbergtrace formula, but in order for this approach to work, the geometric sides of the trace formula for different groupsmust be related in a particular way. This relationship takes the form of identities between orbital integrals onreductive groups G and H over a nonarchimedean local field F, where the group H, called an endoscopic group of G,is constructed from G and some additional data.The first case considered was G = SL2 (Labesse and Langlands, 1979). Langlands and Diana Shelstad then developedthe general framework for the theory of endoscopic transfer and formulated specific conjectures. However, duringthe next two decades only partial progress was made towards proving the fundamental lemma.[4] [5] Thus, it has beencalled a "bottleneck limiting progress on a host of arithmetic questions".[6] Langlands himself, writing on the originsof endoscopy, commented:

“... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of Shimuravarieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and itsendoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemmaand its absence rendered progress almost impossible for more than twenty years.[7] ”

ApproachesA paper of George Lusztig and David Kazhdan pointed out that orbital integrals could be interpreted as countingpoints on certain algebraic varieties over finite fields.[3] Further, the integrals in question can be computed in a waythat depends only on the residue field of F; and the issue can be reduced to the Lie algebra version of the orbitalintegrals. Then the problem was restated in terms of the Springer fiber of algebraic groups.[8] The circle of ideas wasconnected to a purity conjecture; Laumon gave a conditional proof based on such a conjecture, for unitary groups.Laumon and Ngô then introduced the use of the Hitchin fibration, which is an abstract geometric analogue of theHitchin system of mathematical physics, as a further technical tool, leading ultimately to a proof of the "functionfield" case; according to known previous reduction steps, this implies the general form of the conjecture.

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Notes[1] Not even wrong (http:/ / www. math. columbia. edu/ ~woit/ wordpress/ ?p=18)[2] Top 10 Scientific Discoveries of 2009 (http:/ / www. time. com/ time/ specials/ packages/ article/ 0,28804,1945379_1944416_1944435,00.

html), Time[3] The fundamental lemma (http:/ / www. netera. ca/ seminars/ math/ fl-white. pdf), Bill Casselman[4] Kottwitz and Rogawski for U3, Wadspurger for SLn, Hales and Weissauer for Sp4.[5] http:/ / www. newton. ac. uk/ programmes/ ALT/ seminars/ 051316301. pdf[6] http:/ / www. institut. math. jussieu. fr/ projets/ fa/ bpFiles/ Introduction. pdf. p. 1.[7] http:/ / publications. ias. edu/ rpl/ series. php?series=55[8] http:/ / www. claymath. org/ research_award/ Laumon-Ngo/ laumon. pdf, at p. 12.

References• Ngô, Bảo Châu (2008). "Le lemme fondamental pour les algebres de Lie". arXiv:0801.0446..• Laumon, G.; Ngô, B. C. (2004). "Le lemme fondamental pour les groupes unitaires". arXiv:math/0404454..• Dat, Jean-François (Novembre 2004), Lemme fondamental et endoscopie, une approche géométrique, d'après

Gérard Laumon et Ngô Bao Châu (http:/ / www. math. univ-paris13. fr/ ~dat/ publis/ lf. pdf), SéminaireBourbaki, no 940.

• Labesse, J.-P.; Langlands, R. P. (1979), "L-indistinguishability for SL(2)", Can. Jour. Math. 31.

External links• Gerard Laumon lecture on the fundamental lemma for unitary groups (http:/ / www. fields. utoronto. ca/ audio/

02-03/ shimura/ laumon/ )

Fundamental theorem of algebraThe fundamental theorem of algebra states that every non-constant single-variable polynomial with complexcoefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed.Sometimes, this theorem is stated as: every non-zero single-variable polynomial with complex coefficients hasexactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at firstappears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use ofsuccessive polynomial division by linear factors.In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness ofthe reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, itis not fundamental for modern algebra; its name was given at a time in which algebra was mainly about solvingpolynomial equations with real or complex coefficients.

HistoryPeter Rothe (Petrus Roth), in his book Arithmetica Philosophica (published in 1608), wrote that a polynomialequation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle enl'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not statethat they had to be real numbers. Furthermore, he added that his assertion holds “unless the equation is incomplete”,by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clearthat he actually believes that his assertion is always true; for instance, he shows that the equation x4 = 4x − 3,although incomplete, has four solutions (counting multiplicities): 1 (twice), −1 + i√2, and −1 − i√2.

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As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constantpolynomial with real coefficients can be written as a product of polynomials with real coefficients whose degree iseither 1 or 2. However, in 1702 Leibniz said that no polynomial of the type x4 + a4 (with a real and distinct from 0)can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomialx4 − 4x3 + 2x2 + 4x + 4, but he got a letter from Euler in 1742[1] in which he was told that his polynomial happenedto be equal to

where α is the square root of 4 + 2√7. Also, Euler mentioned that

A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Among otherproblems, it assumed implicitly a theorem (now known as Puiseux's theorem) which would not be proved until morethan a century later, and furthermore the proof assumed the fundamental theorem of algebra. Other attempts weremade by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795). These last four attempts assumedimplicitly Girard's assertion; to be more precise, the existence of solutions was assumed and all that remained to beproved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex,Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p(z).At the end of the 18th century, two new proofs were published which did not assume the existence of roots. One ofthem, due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof hadan algebraic gap.[2] The other one was published by Gauss in 1799 and it was mainly geometric, but it had atopological gap, filled by Alexander Ostrowski in 1920, as discussed in Smale 1981 [3] (Smale writes, "...I wish topoint out what an immense gap Gauss' proof contained. It is a subtle point even today that a real algebraic planecurve cannot enter a disk without leaving. In fact even though Gauss redid this proof 50 years later, the gapremained. It was not until 1920 that Gauss' proof was completed. In the reference Gauss, A. Ostrowski has a paperwhich does this and gives an excellent discussion of the problem as well..."). A rigorous proof was published byArgand in 1806; it was here that, for the first time, the fundamental theorem of algebra was stated for polynomialswith complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 1816 and anotherversion of his original proof in 1849.The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique(1821). It contained Argand's proof, although Argand is not credited for it.None of the proofs mentioned so far is constructive. It was Weierstrass who raised for the first time, in the middle ofthe 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra. He presentedhis solution, that amounts in modern terms to a combination of the Durand–Kerner method with the homotopycontinuation principle, in 1891. Another proof of this kind was obtained by Hellmuth Kneser in 1940 and simplifiedby his son Martin Kneser in 1981.Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra forcomplex numbers based on the Dedekind real numbers (which are not constructively equivalent to the Cauchy realnumbers without countable choice[4] ). However, Fred Richman proved a reformulated version of the theorem thatdoes work.[5]

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ProofsAll proofs below involve some analysis, at the very least the concept of continuity of real or complex functions.Some also use differentiable or even analytic functions. This fact has led some to remark that the FundamentalTheorem of Algebra is neither fundamental, nor a theorem of algebra.Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complexroot. This is enough to establish the theorem in the general case because, given a non-constant polynomial p(z) withcomplex coefficients, the polynomial

has only real coefficients and, if z is a zero of q(z), then either z or its conjugate is a root of p(z).A large number of non-algebraic proofs of the theorem use the fact (sometimes called “growth lemma”) that an n-thdegree polynomial function p(z) whose dominant coefficient is 1 behaves like zn when |z| is large enough. A moreprecise statement is: there is some positive real number R such that:

when |z| > R.

Complex-analytic proofsFind a closed disk D of radius r centered at the origin such that |p(z)| > |p(0)| whenever |z| ≥ r. The minimum of |p(z)|on D, which must exist since D is compact, is therefore achieved at some point z0 in the interior of D, but not at anypoint of its boundary. The minimum modulus principle implies then that p(z0) = 0. In other words, z0 is a zero ofp(z).A variation of this proof that does not require the use of the minimum modulus principle (most proofs of which inturn require the use of Cauchy's integral theorem or some of its consequences) is based on the observation that forthe special case of a polynomial function, the minimum modulus principle can be proved directly using elementaryarguments. More precisely, if we assume by contradiction that , then, expanding in powersof we can write

Here, the 's are simply the coefficients of the polynomial , and we let be the index of thefirst coefficient following the constant term that is non-zero. But now we see that for sufficiently close to thishas behavior asymptotically similar to the simpler polynomial , in the sense that (as is

easy to check) the function is bounded by some positive constant in some neighborhood of

. Therefore if we define and let , then for any sufficientlysmall positive number (so that the bound mentioned above holds), using the triangle inequality we see that

When r is sufficiently close to 0 this upper bound for |p(z)| is strictly smaller than |a|, in contradiction to thedefinition of z0. (Geometrically, we have found an explicit direction θ0 such that if one approaches z0 from thatdirection one can obtain values p(z) smaller in absolute value than |p(z0)|.)Another analytic proof can be obtained along this line of thought observing that, since |p(z)| > |p(0)| outside D, the minimum of |p(z)| on the whole complex plane is achieved at z0. If |p(z0)| > 0, then 1/p is a bounded holomorphic function in the entire complex plane since, for each complex number z, |1/p(z)| ≤ |1/p(z0)|. Applying Liouville's

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theorem, which states that a bounded entire function must be constant, this would imply that 1/p is constant andtherefore that p is constant. This gives a contradiction, and hence p(z0) = 0.Yet another analytic proof uses the argument principle. Let R be a positive real number large enough so that everyroot of p(z) has absolute value smaller than R; such a number must exist because every non-constant polynomialfunction of degree n has at most n zeros. For each r > R, consider the number

where c(r) is the circle centered at 0 with radius r oriented counterclockwise; then the argument principle says thatthis number is the number N of zeros of p(z) in the open ball centered at 0 with radius r, which, since r > R, is thetotal number of zeros of p(z). On the other hand, the integral of n/z along c(r) divided by 2πi is equal to n. But thedifference between the two numbers is

The numerator of the rational expression being integrated has degree at most n − 1 and the degree of the denominatoris n + 1. Therefore, the number above tends to 0 as r tends to +∞. But the number is also equal to N − n and so N = n.Still another complex-analytic proof can be given by combining linear algebra with the Cauchy theorem. Toestablish that every complex polynomial of degree n > 0 has a zero, it suffices to show that every complex squarematrix of size n > 0 has a (complex) eigenvalue.[6] The proof of the latter statement is by contradiction.Let A be a complex square matrix of size n > 0 and let In be the unit matrix of the same size. Assume A has noeigenvalues. Consider the resolvent function

which is a meromorphic function on the complex plane with values in the vector space of matrices. The eigenvaluesof A are precisely the poles of R(z). Since, by assumption, A has no eigenvalues, the function R(z) is an entirefunction and Cauchy's theorem implies that

On the other hand, R(z) expanded as a geometric series gives:

This formula is valid outside the closed disc of radius ||A|| (the operator norm of A). Let r > ||A||. Then

(in which only the summand k = 0 has a nonzero integral). This is a contradiction, and so A has an eigenvalue.

Topological proofsLet z0 ∈ C be such that the minimum of |p(z)| on the whole complex plane is achieved at z0; it was seen at the proofwhich uses Liouville's theorem that such a number must exist. We can write p(z) as a polynomial in z − z0: there issome natural number k and there are some complex numbers ck, ck + 1, ..., cn such that ck ≠ 0 and that

It follows that if a is a kth root of −p(z0)/ck and if t is positive and sufficiently small, then |p(z0 + ta)| < |p(z0)|, whichis impossible, since |p(z0)| is the minimum of |p| on D.For another topological proof by contradiction, suppose that p(z) has no zeros. Choose a large positive number R such that, for |z| = R, the leading term zn of p(z) dominates all other terms combined; in other words, such that

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|z|n > |an − 1zn −1 + ··· + a0|. As z traverses the circle given by the equation |z| = R once counter-clockwise, p(z), likezn, winds n times counter-clockwise around 0. At the other extreme, with |z| = 0, the “curve” p(z) is simply the single(nonzero) point p(0), whose winding number is clearly 0. If the loop followed by z is continuously deformedbetween these extremes, the path of p(z) also deforms continuously. We can explicitly write such a deformation as where t is greater than or equal to 0 and less than or equal to 1. If one views the variable t as time, then at time zerothe curve is p(z) and at time one the curve is p(0). Clearly at every point t, p(z) cannot be zero by the originalassumption, therefore during the deformation, the curve never crosses zero. Therefore the winding number of thecurve around zero should never change. However, given that the winding number started as n and ended as 0, this isabsurd. Therefore, p(z) has at least one zero.

Algebraic proofsThese proofs use two facts about real numbers that require only a small amount of analysis (more precisely, theintermediate value theorem):• every polynomial with odd degree and real coefficients has some real root;• every non-negative real number has a square root.The second fact, together with the quadratic formula, implies the theorem for real quadratic polynomials. In otherwords, algebraic proofs of the fundamental theorem actually show that if R is any real-closed field, then its extension

is algebraically closed.As mentioned above, it suffices to check the statement “every non-constant polynomial p(z) with real coefficients hasa complex root”. This statement can be proved by induction on the greatest non-negative integer k such that 2k

divides the degree n of p(z). Let a be the coefficient of zn in p(z) and let F be a splitting field of p(z) over C; in otherwords, the field F contains C and there are elements z1, z2, ..., zn in F such that

If k = 0, then n is odd, and therefore p(z) has a real root. Now, suppose that n = 2km (with m odd and k > 0) and thatthe theorem is already proved when the degree of the polynomial has the form 2k − 1m′ with m′ odd. For a realnumber t, define:

Then the coefficients of qt(z) are symmetric polynomials in the zi's with real coefficients. Therefore, they can beexpressed as polynomials with real coefficients in the elementary symmetric polynomials, that is, in −a1, a2, ...,(−1)nan. So qt(z) has in fact real coefficients. Furthermore, the degree of qt(z) is n(n − 1)/2 = 2k − 1m(n − 1), andm(n − 1) is an odd number. So, using the induction hypothesis, qt has at least one complex root; in other words,zi + zj + tzizj is complex for two distinct elements i and j from {1,...,n}. Since there are more real numbers than pairs(i,j), one can find distinct real numbers t and s such that zi + zj + tzizj and zi + zj + szizj are complex (for the same iand j). So, both zi + zj and zizj are complex numbers. It is easy to check that every complex number has a complexsquare root, thus every complex polynomial of degree 2 has a complex root by the quadratic formula. It follows thatzi and zj are complex numbers, since they are roots of the quadratic polynomial z2 − (zi + zj)z + zizj.J. Shipman showed in 2007 that the assumption that odd degree polynomials have roots is stronger than necessary;any field in which polynomials of prime degree have roots is algebraically closed (so "odd" can be replaced by "oddprime" and furthermore this holds for fields of all characteristics). For axiomatization of algebraically closed fields,this is the best possible, as there are counterexamples if a single prime is excluded. However, these counterexamplesrely on −1 having a square root. If we take a field where −1 has no square root, and every polynomial of degree n ∈ Ihas a root, where I is any fixed infinite set of odd numbers, then every polynomial f(x) of odd degree has a root(since (x2 + 1)kf(x) has a root, where k is chosen so that deg(f) + 2k ∈ I).

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Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to show that C hasno proper finite field extension.[7] Let K/C be a finite extension. Since the normal closure of K over R still has afinite degree over C (or R), we may assume without loss of generality that K is a normal extension of R (hence it is aGalois extension, as every algebraic extension of a field of characteristic 0 is separable). Let G be the Galois groupof this extension, and let H be a Sylow 2-group of G, so that the order of H is a power of 2, and the index of H in Gis odd. By the fundamental theorem of Galois theory, there exists a subextension L of K/R such that Gal(K/L) = H.As [L:R] = [G:H] is odd, and there are no nonlinear irreducible real polynomials of odd degree, we must have L = R,thus [K:R] and [K:C] are powers of 2. Assuming for contradiction [K:C] > 1, the 2-group Gal(K/C) contains asubgroup of index 2, thus there exists a subextension M of C of degree 2. However, C has no extension of degree 2,because every quadratic complex polynomial has a complex root, as mentioned above.

CorollariesSince the fundamental theorem of algebra can be seen as the statement that the field of complex numbers isalgebraically closed, it follows that any theorem concerning algebraically closed fields applies to the field ofcomplex numbers. Here are a few more consequences of the theorem, which are either about the field of realnumbers or about the relationship between the field of real numbers and the field of complex numbers:• The field of complex numbers is the algebraic closure of the field of real numbers.• Every polynomial in one variable x with real coefficients is the product of a constant, polynomials of the form

x + a with a real, and polynomials of the form x2 + ax + b with a and b real and a2 − 4b < 0 (which is the samething as saying that the polynomial x2 + ax + b has no real roots).

• Every rational function in one variable x, with real coefficients, can be written as the sum of a polynomialfunction with rational functions of the form a/(x − b)n (where n is a natural number, and a and b are realnumbers), and rational functions of the form (ax + b)/(x2 + cx + d)n (where n is a natural number, and a, b, c, andd are real numbers such that c2 − 4d < 0). A corollary of this is that every rational function in one variable andreal coefficients has an elementary primitive.

• Every algebraic extension of the real field is isomorphic either to the real field or to the complex field.

Bounds on the zeroes of a polynomialWhile the fundamental theorem of algebra states a general existence result, it is of some interest, both from thetheoretical and from the practical point of view, to have information on the location of the zeroes of a givenpolynomial. The simpler result in this direction is a bound on the modulus: all zeroes of a monic polynomial

satisfy an inequality where

Notice that, as stated, this is not yet an existence result but rather an example of what is called an a priori bound: itsays that if there are solutions then they lie inside the closed disk of center the origin and radius . However, oncecoupled with the fundamental theorem of algebra it says that the disk contains in fact at least one solution. Moregenerally, a bound can be given directly in terms of any p-norm of the n-vector of coefficients ,that is , where is precisely the q-norm of the 2-vector , q being the conjugate exponent of p, 1/p+ 1/q = 1, for any . Thus, the modulus of any solution is also bounded by

for , and in particular

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(where we define to mean 1, which is reasonable since 1 is indeed the n-th coefficient of our polynomial). Thecase of a generic polynomial of degree n, , is of course reduced to the case of amonic, dividing all coefficients by . Also, in case that 0 is not a root, i.e. , bounds from below on theroots follow immediately as bounds from above on , that is, the roots of . Finally,the distance from the roots to any point can be estimated from below and above, seeing aszeroes of the polynomial , whose coefficients are the Taylor expansion of at We report the here the proof of the above bounds, which is short and elementary. Let be a root of the polynomial

; in order to prove the inequality we can assume, of course, . Writing theequation as , and using the Hölder's inequality we find . Now,if , this is , thus . In the case , taking intoaccount the summation formula for a geometric progression, we have

thus and simplifying, . Therefore holds, for all

Notes[1] See section Le rôle d'Euler in C. Gilain's article Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral.[2] Concerning Wood's proof, see the article A forgotten paper on the fundamental theorem of algebra, by Frank Smithies.[3] http:/ / projecteuclid. org/ DPubS?service=UI& version=1. 0& verb=Display& handle=euclid. bams/ 1183547848[4] For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle;

available from (http:/ / www. math. fau. edu/ richman/ HTML/ DOCS. HTM).[5] See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from (http:/ / www.

math. fau. edu/ richman/ HTML/ DOCS. HTM).[6] A proof of the fact that this suffices can be seen here.[7] A proof of the fact that this suffices can be seen here.

References

Historic sources• Cauchy, Augustin Louis (1821), Cours d'Analyse de l'École Royale Polytechnique, 1ère partie: Analyse

Algébrique (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k29058v), Paris: Éditions Jacques Gabay (published 1992),ISBN 2-87647-053-5 (tr. Course on Analysis of the Royal Polytechnic Academy, part 1: Algebraic Analysis)

• Euler, Leonhard (1751), "Recherches sur les racines imaginaires des équations" (http:/ / bibliothek. bbaw. de/bbaw/ bibliothek-digital/ digitalequellen/ schriften/ anzeige/ index_html?band=02-hist/ 1749& seite:int=228),Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin (Berlin) 5: 222–288. Englishtranslation: Euler, Leonhard (1751), "Investigations on the Imaginary Roots of Equations" (http:/ / www.mathsym. org/ euler/ e170. pdf) (PDF), Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin(Berlin) 5: 222–288

• Gauss, Carl Friedrich (1799), Demonstratio nova theorematis omnem functionem algebraicam rationalemintegram unius variabilis in factores reales primi vel secundi gradus resolvi posse, Helmstedt: C. G. Fleckeisen(tr. New proof of the theorem that every integral rational algebraic function of one variable can be resolved intoreal factors of the first or second degree).

• C. F. Gauss, “ Another new proof of the theorem that every integral rational algebraic function of one variable canbe resolved into real factors of the first or second degree (http:/ / www. paultaylor. eu/ misc/ gauss-web. php)”,1815

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• Kneser, Hellmuth (1940), "Der Fundamentalsatz der Algebra und der Intuitionismus" (http:/ / www-gdz. sub.uni-goettingen. de/ cgi-bin/ digbib. cgi?PPN266833020_0046), Mathematische Zeitschrift 46: 287–302,doi:10.1007/BF01181442, ISSN 0025-5874 (The Fundamental Theorem of Algebra and Intuitionism).

• Kneser, Martin (1981), "Ergänzung zu einer Arbeit von Hellmuth Kneser über den Fundamentalsatz der Algebra"(http:/ / www-gdz. sub. uni-goettingen. de/ cgi-bin/ digbib. cgi?PPN266833020_0177), Mathematische Zeitschrift177 (2): 285–287, doi:10.1007/BF01214206, ISSN 0025-5874 (tr. An extension of a work of Hellmuth Kneser onthe Fundamental Theorem of Algebra).

• Ostrowski, Alexander (1920), "Über den ersten und vierten Gaußschen Beweis des Fundamental-Satzes derAlgebra" (http:/ / gdz. sub. uni-goettingen. de/ dms/ load/ img/ ?PPN=PPN236019856& DMDID=dmdlog53),Carl Friedrich Gauss Werke Band X Abt. 2 (tr. On the first and fourth Gaussian proofs of the FundamentalTheorem of Algebra).

• Weierstraß, Karl (1891). "Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichendargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen". Sitzungsberichte derköniglich preussischen Akademie der Wissenschaften zu Berlin. pp. 1085–1101. (tr. New proof of the theoremthat every integral rational function of one variable can be represented as a product of linear functions of the samevariable).

Recent literature• Fine, Benjamin; Rosenber, Gerhard (1997), The Fundamental Theorem of Algebra, Undergraduate Texts in

Mathematics, Berlin: Springer-Verlag, ISBN 978-0-387-94657-3• Gersten, S.M.; Stallings, John R. (1988), "On Gauss's First Proof of the Fundamental Theorem of Algebra",

Proceedings of the AMS 103 (1): 331–332, doi:10.2307/2047574, ISSN 0002-9939, JSTOR 2047574• Gilain, Christian (1991), "Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul

intégral", Archive for History of Exact Sciences 42 (2): 91–136, doi:10.1007/BF00496870, ISSN 0003-9519 (tr.On the history of the fundamental theorem of algebra: theory of equations and integral calculus.)

• Netto, Eugen; Le Vavasseur, Raymond (1916), "Les fonctions rationnelles §80–88: Le théorème fondamental", inMeyer, François; Molk, Jules, Encyclopédie des Sciences Mathématiques Pures et Appliquées, tome I, vol. 2,Éditions Jacques Gabay, 1992, ISBN 2-87647-101-9 (tr. The rational functions §80–88: the fundamentaltheorem).

• Remmert, Reinhold (1991), "The Fundamental Theorem of Algebra", in Ebbinghaus, Heinz-Dieter; Hermes,Hans; Hirzebruch, Friedrich, Numbers, Graduate Texts in Mathematics 123, Berlin: Springer-Verlag,ISBN 978-0-387-97497-2

• Shipman, Joseph (2007), "Improving the Fundamental Theorem of Algebra", Mathematical Intelligencer 29 (4):9–14, doi:10.1007/BF02986170, ISSN 0343-6993

• Smale, Steve (1981), "The Fundamental Theorem of Algebra and Complexity Theory", Bulletin (new series) ofthe American Mathematical Society 4 (1) (http:/ / projecteuclid. org/ DPubS?service=UI& version=1. 0&verb=Display& handle=euclid. bams/ 1183547848)

• Smith, David Eugene (1959), A Source Book in Mathematics, Dover, ISBN 0-486-64690-4• Smithies, Frank (2000), "A forgotten paper on the fundamental theorem of algebra", Notes & Records of the

Royal Society 54 (3): 333–341, doi:10.1098/rsnr.2000.0116, ISSN 0035-9149• van der Waerden, Bartel Leendert (2003), Algebra, I (7th ed.), Springer-Verlag, ISBN 0-387-40624-7

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External links• Fundamental Theorem of Algebra (http:/ / www. cut-the-knot. org/ do_you_know/ fundamental2. shtml) — a

collection of proofs• D. J. Velleman: The Fundamental Theorem of Algebra: A Visual Approach, PDF (unpublished paper) (http:/ /

www. cs. amherst. edu/ ~djv/ ), visualisation of d'Alembert's, Gauss's and the winding number proofs• Fundamental Theorem of Algebra Module by John H. Mathews (http:/ / math. fullerton. edu/ mathews/ c2003/

FunTheoremAlgebraMod. html)• Bibliography for the Fundamental Theorem of Algebra (http:/ / math. fullerton. edu/ mathews/ c2003/

FunTheoremAlgebraBib/ Links/ FunTheoremAlgebraBib_lnk_2. html)• From the Fundamental Theorem of Algebra to Astrophysics: A "Harmonious" Path (http:/ / www. ams. org/

notices/ 200806/ tx080600666p. pdf)

Fundamental theorem of cyclic groupsIn abstract algebra, the fundamental theorem of cyclic groups states that every subgroup of a cyclic group iscyclic. Moreover, the order of any subgroup of a cyclic group of order is a divisor of , and for each positivedivisor of the group has exactly one subgroup of order .

ProofLet be a cyclic group for some and with identity and order , and let be a subgroup of

. We will now show that is cyclic. Ifthen .

If then since is cyclic every element in is of the form , where is an integer. Let be theleast positive integer such that .We will now show that . It follows immediately from the closure property that .To show that

we let . Since we have that for some positive integer .By the division algorithm,

with , and so, which yields .

Now since

and , it follows from closure that .But is the least positive integer such that

and ,which means that and so

.Thus .Since and it follows that and so is cyclic.We will now show that the order of any subgroup of

is a divisor of .Let be any subgroup of of order . We have already shown that

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Fundamental theorem of cyclic groups 108

, where is the least positive integer such that . We know that and therefore we can write

, with .Since

,we must have : , since is the smallest positive integer such that

.It follows that for some integer . Thus .We will now prove the last part of the theorem. Let be any positive divisor of . We will show that

is the one and only subgroup of of order . Note that

has order

.

Let be any subgroup of with order . We know that,

where is a divisor of . So

and .

Consequently

and so ,

and thus the theorem is proved.

Proof by homomorphism with integersLet be a cyclic group, and let be a subgroup of . Define a morphism by

. Since is cyclic generated by , is surjective. Let . is a subgroupof . Since is surjective, the restriction of to defines a surjective homomorphism from onto ,and therefore is isomorphic to a quotient of . Since is a subgroup of , is for some integer . If , then , hence , which is cyclic. Otherwise, is isomorphic to . Therefore isisomorphic to a quotient of , and they are commonly known to be cyclic.

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ConverseThe following statements are equivalent.• A group G of order is cyclic.• For every divisor of a group G has exactly one subgroup of order .• For every divisor of a group G has at most one subgroup of order .

GeneralizationSuppose that R is a right Ore domain in which every left ideal is principal, and let M be a left R-module which isgenerated by n elements. Then each submodule of M can also be generated by n elements (and possibly fewer). Thisresult implies the fundamental theorem of cyclic groups by observing that the ring of integers satisfiesthese conditions, and a cyclic group is precisely a left -module which is generated by one element. (Itssubmodules are its subgroups.)

Fundamental theorem of Galois theoryIn mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types offield extensions.In its most basic form, the theorem asserts that given a field extension E /F which is finite and Galois, there is aone-to-one correspondence between its intermediate fields and subgroups of its Galois group. (Intermediate fields arefields K satisfying F ⊆ K ⊆ E; they are also called subextensions of E /F.)

ProofThe proof of the fundamental theorem is not trivial. The crux in the usual treatment is a rather delicate result of EmilArtin which allows one to control the dimension of the intermediate field fixed by a given group of automorphisms.The automorphisms of a Galois extension K/F are linearly independent as functions over the field K. The proof ofthis fact follows from a more general notion, namely, the linear independence of characters.There is also a fairly simple proof using the primitive element theorem. This proof seems to be ignored by mostmodern treatments, possibly because it requires a separate (but easier) proof in the case of finite fields.[1]

In terms of its abstract structure, there is a Galois connection; most of its properties are fairly formal, but the actualisomorphism of the posets requires some work.

Explicit description of the correspondenceFor finite extensions, the correspondence can be described explicitly as follows.• For any subgroup H of Gal(E /F ), the corresponding field, usually denoted EH, is the set of those elements of E

which are fixed by every automorphism in H.• For any intermediate field K of E /F, the corresponding subgroup is just Aut(E /K ), that is, the set of those

automorphisms in Gal(E /F ) which fix every element of K.For example, the topmost field E corresponds to the trivial subgroup of Gal(E /F ), and the base field F correspondsto the whole group Gal(E /F ).

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Properties of the correspondenceThe correspondence has the following useful properties.• It is inclusion-reversing. The inclusion of subgroups H1 ⊆ H2 holds if and only if the inclusion of fields EH1 ⊇ EH2

holds.• Degrees of extensions are related to orders of groups, in a manner consistent with the inclusion-reversing

property. Specifically, if H is a subgroup of Gal(E /F ), then |H | = [E:EH] and [Gal(E /F ):H ] = [EH:F ].• The field EH is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable

extension is separable) if and only if H is a normal subgroup of Gal(E /F ). In this case, the restriction of theelements of Gal(E /F ) to EH induces an isomorphism between Gal(EH/F ) and the quotient group Gal(E /F )/H.

Example

Lattice of subgroups and subfields

Consider the field K = Q(√2, √3) =Q(√2)(√3). Since K is first determinedby adjoining √2, then √3, each elementof K can be written as:

where a, b, c, d are rational numbers. Its Galois group G = Gal (K/Q) can be determined by examining theautomorphisms of K which fix a. Each such automorphism must send √2 to either √2 or −√2, and must send √3 toeither √3 or −√3 since the permutations in a Galois group can only permute the roots of an irreducible polynomial.Suppose that f exchanges √2 and −√2, so

and g exchanges √3 and −√3, so

These are clearly automorphisms of K. There is also the identity automorphism e which does not change anything,and the composition of f and g which changes the signs on both radicals:

Therefore

and G is isomorphic to the Klein four-group. It has five subgroups, each of which correspond via the theorem to asubfield of K.• The trivial subgroup (containing only the identity element) corresponds to all of K.• The entire group G corresponds to the base field Q.• The two-element subgroup {1, f } corresponds to the subfield Q(√3), since f fixes √3.• The two-element subgroup {1, g} corresponds to the subfield Q(√2), again since g fixes √2.• The two-element subgroup {1, fg} corresponds to the subfield Q(√6), since fg fixes √6.

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Example

Lattice of subgroups and subfields

The following is the simplest casewhere the Galois group is not abelian.Consider the splitting field K of thepolynomial x3−2 over Q; that is, K = Q(θ, ω), where θ is a cube root of 2, andω is a cube root of 1 (but not 1 itself).For example, if we imagine K to beinside the field of complex numbers,we may take θ to be the real cube rootof 2, and ω to be

It can be shown that the Galois group G = Gal (K/Q) has six elements, and is isomorphic to the group ofpermutations of three objects. It is generated by (for example) two automorphisms, say f and g, which are determinedby their effect on θ and ω,

and then

The subgroups of G and corresponding subfields are as follows:• As usual, the entire group G corresponds to the base field Q, and the trivial group {1} corresponds to the whole

field K.• There is a unique subgroup of order 3, namely {1, f, f 2}. The corresponding subfield is Q(ω), which has degree

two over Q (the minimal polynomial of ω is x2 + x + 1), corresponding to the fact that the subgroup has index twoin G. Also, this subgroup is normal, corresponding to the fact that the subfield is normal over Q.

• There are three subgroups of order 2, namely {1, g}, {1, gf } and {1, gf 2}, corresponding respectively to the threesubfields Q(θ), Q(ωθ), Q(ω2θ). These subfields have degree three over Q, again corresponding to the subgroupshaving index 3 in G. Note that the subgroups are not normal in G, and this corresponds to the fact that thesubfields are not Galois over Q. For example, Q(θ) contains only a single root of the polynomial x3−2, so itcannot be normal over Q.

ApplicationsThe theorem converts the difficult-sounding problem of classifying the intermediate fields of E /F into the moretractable problem of listing the subgroups of a certain finite group.For example, to prove that the general quintic equation is not solvable by radicals (see Abel–Ruffini theorem), onefirst restates the problem in terms of radical extensions (extensions of the form F(α) where α is an n-th root of someelement of F), and then uses the fundamental theorem to convert this statement into a problem about groups. Thatcan then be attacked directly.Theories such as Kummer theory and class field theory are predicated on the fundamental theorem.

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Infinite caseThere is also a version of the fundamental theorem that applies to infinite algebraic extensions, which are normal andseparable. It involves defining a certain topological structure, the Krull topology, on the Galois group; onlysubgroups that are also closed sets are relevant in the correspondence.

References[1] See Marcus, Daniel (1977). Number Fields. Appendix 2. New York: Springer-Verlag. ISBN 0387902791.

Fundamental theorem of linear algebraIn mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces.These may be stated concretely in terms of the rank r of an m×n matrix A and its singular value decomposition:

First, each matrix ( has rows and columns) induces four fundamental subspaces. Thesefundamental subspaces are:

name of subspace definition containing space dimension basis

column space, range or image or (rank) The first columns of

nullspace or kernel or (nullity) The last columns of

row space or coimage or The first rows of

left nullspace or cokernel or The last rows of

Secondly:

1. In , , that is, the nullspace is the orthogonal complement of the row space2. In , , that is, the left nullspace is the orthogonal complement of the column space.

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Fundamental theorem of linear algebra 113

The four subspaces associated to a matrix A.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewritesthis in terms of abstract vector spaces, operators, and the dual spaces as and : thekernel and image of are the cokernel and coimage of .

References• Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.• Strang, Gilbert (1993), "The fundamental theorem of linear algebra" [1], American Mathematical Monthly 100 (9):

848–855, doi:10.2307/2324660, JSTOR 2324660

External links• Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces [2] at Google Video, from MIT

OpenCourseWare

References[1] http:/ / www. eng. iastate. edu/ ~julied/ classes/ CE570/ Notes/ strangpaper. pdf[2] http:/ / ocw. mit. edu/ OcwWeb/ Mathematics/ 18-06Spring-2005/ VideoLectures/ detail/ lecture10. htm

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Fundamental theorem on homomorphismsIn abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamentalhomomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of thekernel and image of the homomorphism.The homomorphism theorem is used to prove the isomorphism theorems.

Group theoretic versionGiven two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the naturalsurjective homomorphism G→G/K. If K ⊂ ker(f) then there exists a unique homomorphism h:G/K→H such that f = hφ.The situation is described by the following commutative diagram:

By setting K = ker(f) we immediately get the first isomorphism theorem.

Other versionsSimilar theorems are valid for monoids, vector spaces, modules, and rings.

External links• A proof at planetmath [1]

References[1] http:/ / planetmath. org/ encyclopedia/ FundamentalHomomorphismTheorem. html

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GilmanGriess theorem 115

Gilman–Griess theoremIn finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by (Gilman & Griess 1983),classifies the finite simple groups of characteristic 2 type with e(G) ≥ 4 that have a "standard component", whichcovers one of the three cases of the trichotomy theorem.

References• Gilman, Robert H.; Griess, Robert L. (1983), "Finite groups with standard components of Lie type over fields of

characteristic two", Journal of Algebra 80 (2): 383–516, doi:10.1016/0021-8693(83)90007-8, ISSN 0021-8693,MR691810

Going up and going downIn commutative algebra, a branch of mathematics, going up and going down are terms which refer to certainproperties of chains of prime ideals in integral extensions.The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going downrefers to the case when a chain can be extended by "downward inclusion".The major results are the Cohen–Seidenberg theorems, which were proved by Irving S. Cohen and AbrahamSeidenberg. These are colloquially known as the going-up and going-down theorems.

Going up and going downLet A⊆B be an extension of commutative rings.The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in B, each member ofwhich lies over members of a longer chain of prime ideals in A, can be extended to the length of the chain of primeideals in A.

Lying over and incomparabilityFirst, we fix some terminology. If and are prime ideals of A and B, respectively, such that

then we say that lies under and that lies over . In general, a ring extension A⊆B of commutative rings issaid to satisfy the lying over property if every prime ideal P of A lies under some prime ideal Q of B.The extension A⊆B is said to satisfy the incomparability property if whenever Q and Q' are distinct primes of Blying over prime P in A, then Q⊈Q' and Q' ⊈Q.

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Going-upThe ring extension A⊆B is said to satisfy the going-up property if whenever

is a chain of prime ideals of A and

(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m, lies over , then the chain

can be extended to a chain

such that for each 1 ≤ i ≤ n, lies over .In (Kaplansky 1970) it is shown that if an extension A⊆B satisfies the going-up property, then it also satsifies thelying-over property.

Going downThe ring extension A⊆B is said to satisfy the going-down property if whenever

is a chain of prime ideals of A and

(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m, lies over , then the chain

can be extended to a chain

such that for each 1 ≤ i ≤ n, lies over .There is a generalization of the ring extension case with ring morphisms. Let f : A → B be a (unital) ringhomomorphism so that B is a ring extension of f(A). Then f is said to satisfy the going-up property if the going-upproperty holds for f(A) in B.Similarly, if f(A) is a ring extension, then f is said to satisfy the going-down property if the going-down propertyholds for f(A) in B.In the case of ordinary ring extensions such as A⊆B, the inclusion map is the pertinent map.

Going-up and going-down theoremsThe usual statements of going-up and going-down theorems refer to a ring extension A⊆B:1. (Going up) If B is an integral extension of A, then the extension satisfies the going-up property (and hence the

lying over property), and the incomparability property.2. (Going down) If B is an integral extension of A, and B is a domain, and A is integrally closed in its field of

fractions, then the extension (in addition to going-up, lying-over and incomparability) satisfies the going-downproperty.

There is another sufficient condition for the going-down property:• If A⊆B is a flat extension of commutative rings, then the going-down property holds[1] .Proof:[2] Let p1⊆p2 be prime ideals of A and let q2 be a prime ideal of B such that q2 ∩ A = p2. We wish to prove that there is a prime ideal q1 of B contained in q2 such that q1 ∩ A = p1. Since A⊆B is a flat extension of rings, it follows

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Going up and going down 117

that Ap2⊆Bq2 is a flat extension of rings. In fact, Ap2⊆Bq2 is a faithfully flat extension of rings since the inclusionmap Ap2 → Bq2 is a local homomorphism. Therefore, the induced map on spectra Spec(Bq2) → Spec(Ap2) issurjective and there exists a prime ideal of Bq2 that contracts to the prime ideal p1Ap2 of Ap2. The contraction of thisprime ideal of Bq2 to B is a prime ideal q1 of B contained in q2 that contracts to p1. The proof is complete. Q.E.D.

References[1] This follows from a much more general lemma in Bruns-Herzog, Lemma A.9 on page 415.[2] Matsumura, page 33, (5.D), Theorem 4

• Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN0-201-00361-9 MR242802

• Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39.Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1

• Kaplansky, Irving, Commutative rings, Allyn and Bacon, 1970.• Matsumura, Hideyuki (1970). Commutative algebra. W. A. Benjamin. ISBN 978-0805370256.• Sharp, R. Y. (2000). "13 Integral dependence on subrings (13.38 The going-up theorem, pp. 258–259; 13.41 The

going down theorem, pp. 261–262)". Steps in commutative algebra. London Mathematical Society Student Texts.51 (Second ed.). Cambridge: Cambridge University Press. pp. xii+355. ISBN 0-521-64623-5. MR1817605.

Goldie's theoremIn mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (="finite rank") as aright module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie'stheorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian rightclassical ring of quotients. The structure of this ring of quotients is then completely determined by theArtin–Wedderburn theorem.In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian ringshave the ascending chain condition on all right ideals. This is sufficient to guarantee that a ring is right Goldie.

References• Coutinho, S.C. & J.C. McConnell (2003) "The quest for quotient rings (of non-commutative Noetherian rings)",

American Mathematical Monthly 110: 298–313.• Goldie, A.W. (1958). "The structure of prime rings under ascending chain conditions". Proc. London Math. Soc. 8

(4): 589–608. doi:10.1112/plms/s3-8.4.589.• Goldie, A.W. (1960). "Semi-prime rings with maximal conditions". Proc. London Math. Soc. 10: 201–220.

doi:10.1112/plms/s3-10.1.201.• Herstein, I.N. (1969). Topics in ring theory. Chicago lectures in mathematics. Chicago, Ill.: Chicago Univ. Pr..

pp. 61–86. ISBN 0-226-32802-3.

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External links• PlanetMath page on Goldie's theorem [1]

• PlanetMath page on Goldie ring [2]

References[1] http:/ / planetmath. org/ encyclopedia/ GoldiesTheorem. html[2] http:/ / planetmath. org/ encyclopedia/ GoldieRing. html

Golod–Shafarevich theoremIn mathematics, the Golod–Shafarevich theorem was proved in 1964 by two Russian mathematicians, EvgenyGolod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences invarious branches of algebra.

The inequalityLet A = K<x1, ..., xn> be the free algebra over a field K in n = d + 1 non-commuting variables xi.Let J be the 2-sided ideal of A generated by homogeneous elements fj of A of degree dj with

2 ≤ d1 ≤ d2 ≤ ...where dj tends to infinity. Let ri be the number of dj equal to i.Let B=A/J, a graded algebra. Let bj = dim Bj.The fundamental inequality of Golod and Shafarevich states that

As a consequence:• B is infinite-dimensional if ri ≤ d2/4 for all i• if B is finite-dimensional, then ri > d2/4 for some i.

ApplicationsThis result has important applications in combinatorial group theory:• If G is a nontrivial finite p-group, then r > d2/4 where d = dim H1(G,Z/pZ) and r = dim H2(G,Z/pZ) (the mod p

cohomology groups of G). In particular if G is a finite p-group with minimal number of generators d and has rrelators in a given presentation, then r > d2/4.

• For each prime p, there is an infinite group G generated by three elements in which each element has order apower of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitelygenerated infinite torsion group, although there is no uniform bound on the order of its elements.

In class field theory, the class field tower of a number field K is created by iterating the Hilbert class fieldconstruction. Another consequence of the construction is that such towers may be infinite (in other words, do notalways terminate in a field equal to its Hilbert class field).

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References1. Golod, E.S; Shafarevich, I.R. (1964), "On the class field tower", Izv. Akad. Nauk SSSSR 28: 261–272 (in Russian)

MR01618522. Golod, E.S (1964), "On nil-algebras and finitely approximable p-groups.", Izv. Akad. Nauk SSSSR 28: 273–276

(in Russian) MR01618783. Herstein, I.N. (1968), "Noncommutative rings," Carus Mathematical Monographs, MAA. ISBN 0-88385-039-7.

See Chapter 8.4. Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press.

ISBN 0-521-23108-6. See chapter VI.5. Roquette, P. (1967), On class field towers,pages 231–249 in Algebraic number theory, Proceedings of the

instructional conference held at the University of Sussex, Brighton, September 1–17 , 1965. Edited by J. W. S.Cassels and A. Fröhlich. Reprint of the 1967 original. Academic Press, London, 1986. xviii+366 pp. ISBN0-12-163251-2

6. Serre, J.-P. (2002), "Galois Cohomology," Springer-Verlag. ISBN 3-540-42192-0. See Appendix 2. (Translationof Cohomologie Galoisienne, Lecture Notes in Mathematics 5, 1973.)

Gorenstein–Harada theoremIn mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada (1973,1974) in a 464 page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part of theclassification of finite simple groups.Finite simple groups of section 2 rank at least 5 have Sylow 2-subgroups with a self-centralizing normal subgroup ofrank at least 3, which implies that they have to be of either component type or of characteristic 2 type. Therefore theGorenstein–Harada theorem splits the problem of classifying finite simple groups into these two subcases.

References• Gorenstein, D.; Harada, Koichiro (1973), "Finite groups of sectional 2-rank at most 4", in Gagen, Terrence; Hale,

Mark P. Jr.; Shult, Ernest E., Finite groups '72. Proceedings of the Gainesville Conference on Finite Groups,March 23-24, 1972, North-Holland Math. Studies, 7, Amsterdam: North-Holland, pp. 57–67,ISBN 978-0-444-10451-9, MR0352243

• Gorenstein, D.; Harada, Koichiro (1974), Finite groups whose 2-subgroups are generated by at most 4 elements[1], Memoirs of the American Mathematical Society, 147, Providence, R.I.: American Mathematical Society,ISBN 978-0-8218-1847-3, MR0367048

References[1] http:/ / books. google. com/ books?id=CzUZAQAAIAAJ

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Gromov's theorem on groups of polynomialgrowthIn geometric group theory, Gromov's theorem on groups of polynomial growth, named for Mikhail Gromov,characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups offinite index.The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated grouphas polynomial growth means the number of elements of length (relative to a symmetric generating set) at most n isbounded above by a polynomial function p(n). The order of growth is then the least degree of any such polynomialfunction p.A nilpotent group G is a group with a lower central series terminating in the identity subgroup.Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotentsubgroup that is of finite index.There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of Joseph A. Wolfshowed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Yves Guivarc'h andindependently Hyman Bass (with different proofs) computed the exact order of polynomial growth. Let G be afinitely generated nilpotent group with lower central series

In particular, the quotient group Gk/Gk+1 is a finitely generated abelian group.The Bass–Guivarch formula states that the order of polynomial growth of G is

where:rank denotes the rank of an abelian group, i.e. the largest number of independent and torsion-free elements ofthe abelian group.

In particular, Gromov's theorem and the Bass–Guivarch formula imply that the order of polynomial growth of afinitely generated group is always either an integer or infinity (excluding for example, fractional powers).In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now calledthe Gromov–Hausdorff convergence, is currently widely used in geometry.A relatively simple proof of the theorem was found by Bruce Kleiner. Later, Terence Tao and Yehuda Shalommodified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicitbounds.[1] [2]

References[1] http:/ / terrytao. wordpress. com/ 2010/ 02/ 18/ a-proof-of-gromovs-theorem/[2] Yehuda Shalom; Terence Tao (2009). "A finitary version of Gromov's polynomial growth theorem". arXiv:0910.4148 [math.GR].

• H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proceedings LondonMathematical Society, vol 25(4), 1972

• M. Gromov, Groups of Polynomial growth and Expanding Maps, Publications mathematiques I.H.É.S., 53, 1981(http:/ / www. numdam. org/ numdam-bin/ feuilleter?id=PMIHES_1981__53_)

• Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A–B 272 (1971). (http:/ /www. numdam. org/ item?id=BSMF_1973__101__333_0)

• Kleiner, Bruce (2007). "A new proof of Gromov's theorem on groups of polynomial growth". arXiv:0710.4593.

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• J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, Journal ofDifferential Geometry, vol 2, 1968

Grushko theoremIn the mathematical subject of group theory, the Grushko theorem or the Grushko-Neumann theorem is atheorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups isequal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko[1]

and then, independently, in a 1943 article of Neumann.[2]

Statement of the theoremLet A and B be finitely generated groups and let A∗B be the free product of A and B. Then

rank(A∗B) = rank(A) + rank(B).It is obvious that rank(A∗B) ≤ rank(A) + rank(B) since if X is a finite generating set of A and Y is a finite generatingset of B then X∪Y is a generating set for A∗B and that |X∪Y|≤|X| + |Y|. The opposite inequality, rank(A∗B) ≥ rank(A)+ rank(B), requires proof.There is a more precise version of Grushko's theorem in terms of Nielsen equivalence. It states that if M = (g1, g2, ...,gn) is an n-tuple of elements of G = A∗B such that M generates G, <g1, g2, ..., gn> = G, then M is Nielsen equivalentin G to an n-tuple of the form

M' = (a1, ..., ak, b1, ..., bn−k) where {a1, ..., ak}⊆A is a generating set for A and where {b1, ..., bn−k}⊆B is agenerating set for B. In particular, rank(A) ≤ k, rank(B) ≤ n − k and rank(A) + rank(B) ≤ k + (n − k) = n. If onetakes M to be the minimal generating tuple for G, that is, with n = rank(G), this implies that rank(A) + rank(B)≤ rank(G). Since the opposite inequality, rank(G) ≤ rank(A) + rank(B), is obvious, it follows thatrank(G)=rank(A) + rank(B), as required.

History and generalizationsAfter the original proofs of Grushko (1940) and Neumann(1943), there were many subsequent alternative proofs,simplifications and generalizations of Grushko's theorem. A close version of Grushko's original proof is given in the1955 book of Kurosh.[3]

Like the original proofs, Lyndon's proof (1965)[4] relied on length-functions considerations but with substantialsimplifications. A 1965 paper of Stallings [5] gave a greatly simplified topological proof of Grushko's theorem.A 1970 paper of Zieschang[6] gave a Nielsen equivalence version of Grushko's theorem (stated above) and providedsome generalizations of Grushko's theorem for amalgamated free products. Scott (1974) gave another topologicalproof of Grushko's theorem, inspired by the methods of 3-manifold topology[7] Imrich (1984)[8] gave a version ofGrushko's theorem for free products with infinitely many factors.Modern techniques of Bass-Serre theory, particularly the machinery of foldings for group actions on trees and forgraphs of groups provide a relatively straightforward proof of Grushko's theorem (see, for example [9] [10] ).Grushko's theorem is, in a sense, a starting point in Dunwoody's theory of accessibility for finitely generated andfinitely presented groups. Since the ranks of the free factors are smaller than the rank of a free product, Grushko'stheorem implies that the process of iterated splitting of a finitely generated group G as a free product must terminatein a finite number of steps (more precisely, in at most rank(G) steps). There is a natural similar question for iteratingsplittings of finitely generated groups over finite subgroups. Dunwoody proved that such a process must alwaysterminate if a group G is finitely presented[11] but may go on forever if G is finitely generated but not finitelypresented.[12]

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An algebraic proof of a substantial generalization of Grushko's theorem using the machinery of groupoids was givenby Higgins (1966).[13] Higgins' theorem starts with groups G and B with free decompositions G = ∗i Gi, B = ∗i Bi andf : G → B a morphism such that f(Gi) = Bi for all i. Let H be a subgroup of G such that f(H) = B. Then H has adecomposition H = ∗i Hi such that f(Hi) = Bi for all i. Full details of the proof and applications may also be found in.[14]

Grushko decomposition theoremA useful consequence of the original Grushko theorem is the so-called Grushko decomposition theorem. It assertsthat any nontrivial finitely generated group G can be decomposed as a free product

G = A1∗A2∗...∗Ar∗Fs, where s ≥ 0, r ≥ 0,where each of the groups Ai is nontrivial, freely indecomposable (that is, it cannot be decomposed as a free product)and not infinite cyclic, and where Fs is a free group of rank s; moreover, for a given G, the groups A1, ..., Ar areunique up to a permutation of their conjugacy classes in G (and, in particular, the sequence of isomorphism types ofthese groups is unique up to a permutation) and the numbers s and r are unique as well.More precisely, if G = B1∗...∗Bk∗Ft is another such decomposition then k = r, s = t, and there exists a permutationσ∈Sr such that for each i=1,...,r the subgroups Ai and Bσ(i) are conjugate in G.The existence of the above decomposition, called the Grushko decomposition of G, is an immediate corollary of theoriginal Grushko theorem, while the uniqueness statement requires additional arguments (see, for example[15] ).Algorithmically computing the Grushko decomposition for specific classes of groups is a difficult problem whichprimarily requires being able to determine if a given group is freely decomposable. Positive results are available forsome classes of groups such as torsion-free word-hyperbolic groups, certain classes of relatively hyperbolicgroups,[16] fundamental groups of finite graphs of finitely generated free groups[17] and others.Grushko decomposition theorem is a group-theoretic analog of the Kneser prime decomposition theorem for3-manifolds which says that a closed 3-manifold can be uniquely decomposed as a connected sum of irreducible3-manifolds.[18]

Sketch of the proof using Bass-Serre theoryThe following is a sketch of the proof of Grushko's theorem based on the use of foldings techniques for groupsacting on trees (see [9] [10] for complete proofs using this argument).Let S={g1,....,gn} be a finite generating set for G=A∗B of size |S|=n=rank(G). Realize G as the fundamental group ofa graph of groups Y which is a single non-loop edge with vertex groups A and B and with the trivial edge group. Let

be the Bass-Serre covering tree for Y. Let F=F(x1,....,xn) be the free group with free basis x1,....,xn and let φ0:F →G be the homomorphism such that φ0(xi)=gi for i=1,...,n. Realize F as the fundamental group of a graph Z0 which isthe wedge of n circles that correspond to the elements x1,....,xn. We also think of Z0 as a graph of groups with theunderlying graph Z0 and the trivial vertex and edge groups. Then the universal cover of Z0 and the Bass-Serrecovering tree for Z0 coincide. Consider a φ0-equivariant map so that it sends vertices to vertices andedges to edge-paths. This map is non-injective and, since both the source and the target of the map are trees, this map"folds" some edge-pairs in the source. The graph of groups Z0 serves as an initial approximation for Y.We now start performing a sequence of "folding moves" on Z0 (and on its Bass-Serre covering tree) to construct asequence of graphs of groups Z0, Z1, Z2, ...., that form better and better approximations for Y. Each of the graphs ofgroups Zj has trivial edge groups and comes with the following additional structure: for each nontrivial vertex groupof it there assigned a finite generating set of that vertex group. The complexity c(Zj) of Zj is the sum of the sizes ofthe generating sets of its vertex groups and the rank of the free group π1(Zj). For the initial approximation graph wehave c(Z0)=n.

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The folding moves that take Zj to Zj+1 can be of one of two types:• folds that identify two edges of the underlying graph with a common initial vertex but distinct end-vertices into a

single edge; when such a fold is performed, the generating sets of the vertex groups and the terminal edges are"joined" together into a generating set of the new vertex group; the rank of the fundamental group of theunderlying graph does not change under such a move.

• folds that identify two edges, that already had common initial vertices and common terminal vertices, into a singleedge; such a move decreases the rank of the fundamental group of the underlying graph by 1 and an element thatcorresponded to the loop in the graph that is being collapsed is "added" to the generating set of one of the vertexgroups.

One sees that the folding moves do not increase complexity but they do decrease the number of edges in Zj.Therefore the folding process must terminate in a finite number of steps with a graph of groups Zk that cannot befolded any more. It follows from the basic Bass-Serre theory considerations that Zk must in fact be equal to the edgeof groups Y and that Zk comes equipped with finite generating sets for the vertex groups A and B. The sum of thesizes of these generating sets is the complexity of Zk which is therefore less than or equal to c(Z0)=n. This impliesthat the sum of the ranks of the vertex groups A and B is at most n, that is rank(A)+rank(B)≤rank(G), as required.

Notes[1] I. A. Grushko, On the bases of a free product of groups, Matematicheskii Sbornik, vol 8 (1940), pp. 169–182.[2] B. H. Neumann. On the number of generators of a free product. Journal of the London Mathematical Society, vol 18, (1943), pp. 12–20.[3] A. G. Kurosh, The theory of groups. Vol. I. Translated and edited by K. A. Hirsch. Chelsea Publishing Co., New York, N.Y., 1955[4] , Roger C. Lyndon, Grushko's theorem. Proceedings of the American Mathematical Society, vol. 16 (1965), pp. 822–826.[5] John R. Stallings. A topological proof of Grushko's theorem on free products. Mathematische Zeitschrift, vol. 90 (1965), pp. 1–8.[6] Heiner Zieschang. Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam. Inventiones Mathematicae, vol. 10 (1970), pp.

4–37[7] Scott, Peter. An introduction to 3-manifolds. Department of Mathematics, University of Maryland, Lecture Note, No. 11. Department of

Mathematics, University of Maryland, College Park, Maryland, 1974[8] Wilfried Imrich Grushko's theorem. Archiv der Mathematik (Basel), vol. 43 (1984), no. 5, pp. 385-387[9] John R. Stallings. Foldings of G-trees. Arboreal group theory (Berkeley, California, 1988), pp. 355–368, Mathematical Sciences Research

Institute Publications, 19. Springer, New York, 1991; ISBN 0-387-97518-7[10] Ilya Kapovich, Richard Weidmann, and Alexei Miasnikov. Foldings, graphs of groups and the membership problem. International Journal

of Algebra and Computation, vol. 15 (2005), no. 1, pp. 95–128[11] Martin J. Dunwoody. The accessibility of finitely presented groups. Inventiones Mathematicae, vol. 81 (1985), no. 3, pp. 449–457[12] Martin J. Dunwoody. An inaccessible group. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 75–78, London Mathematical Society

Lecture Notes Series, 181, Cambridge University Press, Cambridge, 1993. ISBN 0-521-43529-3[13] P. J. Higgins. Grushko's theorem. Journal of Algebra, vol. 4 (1966), pp. 365–372[14] Higgins, Philip J., Notes on categories and groupoids. Van Nostrand Rienhold Mathematical Studies, No. 32. Van Nostrand Reinhold Co.,

London-New York-Melbourne, 1971. Reprinted as Theory and Applications of Categories Reprint No 7, 2005. (http:/ / www. tac. mta. ca/ tac/reprints/ articles/ 7/ tr7abs. html)

[15] John Stallings. Coherence of 3-manifold fundamental groups. (http:/ / www. numdam. org/ numdam-bin/fitem?id=SB_1975-1976__18__167_0) Séminaire Bourbaki, 18 (1975-1976), Exposé No. 481.

[16] François Dahmani and Daniel Groves. Detecting free splittings in relatively hyperbolic groups. (http:/ / www. ams. org/ tran/ 0000-000-00/S0002-9947-08-04486-3/ ) Transactions of the American Mathematical Society. Posted online July 21, 2008.

[17] Guo-An Diao and Mark Feighn. The Grushko decomposition of a finite graph of finite rank free groups: an algorithm. (http:/ / msp.warwick. ac. uk/ gt/ 2005/ 09/ p041. xhtml) Geometry and Topology. vol. 9 (2005), pp. 1835–1880

[18] H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresber. Deutsch. Math. Verein., vol. 38 (1929), pp. 248–260

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Haboush's theoremIn mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for anysemisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, givenv ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V such that

F(v) ≠ 0.The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V,and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p. When K hascharacteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of Gimplies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p wasproved by W. J. Haboush (1975), about a decade after the problem had been posed by David Mumford, in theintroduction to the first edition of his book Geometric Invariant Theory.

ApplicationsHaboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where theywere already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theoremshow that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixedsubalgebra is also finitely generated.Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety,then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and1 on Y).C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.It follows from the work of Nagata (1963), Haboush, and Popov that the following conditions are equivalent for anaffine algebraic group G over a field K:• G is reductive (its unipotent radical is trivial).• For any non-zero invariant vector in a rational representation on G, there is an invariant homogeneous polynomial

that does not vanish on it.• For any finitely generated K algebra acted on rationally by G, the algebra of fixed elements is finitely generated.

ProofThe theorem is proved in several steps as follows:• We can assume that the group is defined over an algebraically closed field K of characteristic p>0.• Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the case

of connected reductive groups (as the connected component has finite index). By taking a central extensionwhich is harmless one can also assume the group G is simply connected.

• Let A(G) be the coordinate ring of G. This is a representation of G with G acting by left translations. Pick anelement v′ of the dual of V that has value 1 on the invariant vector v. The map V to A(G) by sending w∈V to theelement a∈A(G) with a(g) = v′(g(w)). This sends v to 1∈A(G), so we can assume that V⊂A(G) and v=1.

• The structure of the representation A(G) is given as follows. Pick a maximal torus T of G, and let it act on A(G) byright translations (so that it commutes with the action of G). Then A(G) splits as a sum over characters λ of T ofthe subrepresentations A(G)λ of elements transforming according to λ. So we can assume that V is contained in theT-invariant subspace A(G)λ of A(G).

• The representation A(G)λ is an increasing union of subrepresentations of the form Eλ+nρ⊗Enρ, where ρ is the Weyl vector for a choice of simple roots of T, n is a positive integer, and Eμ is the space of sections of the line

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bundle over G/B corresponding to a character μ of T, where B is a Borel subgroup containing T.• If n is sufficiently large then Enρ has dimension (n+1)N where N is the number of positive roots. This is because in

characteristic 0 the corresponding module has this dimension by the Weyl character formula, and for n largeenough that the line bundle over G/B is very ample, Enρ has the same dimension as in characteristic 0.

• If q=pr for a positive integer r, and n=q−1, then Enρ contains the Steinberg representation of G(Fq) of dimensionqN. (Here Fq ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation ofG(Fq) and therefore of G(K), and for r large enough it has the same dimension as Enρ, so there are infinitely manyvalues of n such that Enρ is irreducible.

• If Enρ is irreducible it is isomorphic to its dual, so Enρ⊗Enρ is isomorphic to End(Enρ). Therefore the T-invariantsubspace A(G)λ of A(G) is an increasing union of subrepresentations of the form End(E) for representations E (ofthe form E(q−1)ρ)). However for representations of the form End(E) an invariant polynomial that separates 0 and 1is given by the determinant. This completes the sketch of the proof of Haboush's theorem.

References• Demazure, Michel (1976), "Démonstration de la conjecture de Mumford (d'après W. Haboush)", Séminaire

Bourbaki (1974/1975: Exposés Nos. 453--470), Lecture Notes in Math., 514, Berlin: Springer, pp. 138–144,doi:10.1007/BFb0080063, ISBN 978-3-540-07686-5, MR0444786

• Haboush, W. J. (1975), "Reductive groups are geometrically reductive", Ann. Of Math. (The Annals ofMathematics, Vol. 102, No. 1) 102 (1): 67–83, doi:10.2307/1970974, JSTOR 1970974

• Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik undihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.xiv+292 pp. MR1304906 ISBN 3-540-56963-4

• Nagata, Masayoshi (1963), "Invariants of a group in an affine ring" [1], Journal of Mathematics of KyotoUniversity 3: 369–377, ISSN 0023-608X, MR0179268

• M. Nagata, T. Miyata, "Note on semi-reductive groups" J. Math. Kyoto Univ. , 3 (1964) pp. 379–382• Popov, V.L. (2001), "Mumford hypothesis" [2], in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer,

ISBN 978-1556080104• C.S. Seshadri, "Geometric reductivity over arbitrary base" Adv. Math. , 26 (1977) pp. 225–274

References[1] http:/ / projecteuclid. org/ euclid. kjm/ 1250524787[2] http:/ / eom. springer. de/ M/ m065570. htm

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Hahn embedding theorem 126

Hahn embedding theoremIn mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, theHahn embedding theorem gives a simple description of all linearly ordered abelian groups.The theorem states: Any linearly ordered abelian group can be embedded as an ordered subgroup of the additivegroup ℝΩ endowed with a lexicographical order, where ℝ is the additive group of real numbers (with its standardorder), and Ω is the set of Archimedean equivalence classes of .

Let denote the identity element of . For any nonzero ∈ , exactly one of the elements or isgreater than ; denote this element by . Two nonzero elements ∈ are Archimedean equivalent if thereexist natural numbers ∈ℕ such that and . (Heuristically: neither nor is"infinitesimal" with respect to the other). The group is Archimedean if all nonzero elements areArchimedean-equivalent. In this case, Ω is a singleton, so ℝΩ is just the group of real numbers. Then Hahn'sEmbedding Theorem reduces to Hölder's theorem (which states that a linearly ordered abelian group is Archimedeanif and only if it is a subgroup of the ordered additive group of the real numbers).(Gravett 1956) gives a clear statement and proof of the theorem. The papers of (Clifford 1954) and (Hausner &Wendel 1952) together provide another proof. See also (Fuchs & Salce 2001, p. 62).

References• Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and

Monographs, 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR1794715• Hahn, H. (1907), "Über die nichtarchimedischen Größensysteme." (in German), Sitzungsberichte der

Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse (Wien. Ber.)116: 601–655

• Gravett, K. A. H. (1956), "Ordered Abelian Groups", Quarterly Journal of Mathematics of Oxford Series 2 7:57–63, doi:10.1093/qmath/7.1.57

• Clifford, A.H. (1954), "Note on Hahn's Theorem on Ordered Abelian Groups", Proceedings of the AmericanMathematical Society 5 (6): 860–863

• Hausner, M.; Wendel, J.G. (1952), "Ordered vector spaces", Proceedings of the American Mathematical Society3: 977–982, doi:10.1090/S0002-9939-1952-0052045-1

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Hajós's theorem 127

Hajós's theoremIn group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product ofsimplexes, that is, sets of the form {e,a,a2,...,as-1} where e is the identity element, then at least one of the factors is asubgroup. The theorem was proved by the Hungarian mathematician György Hajós in 1941 using group rings. Rédeilater proved the statement when the factors are only required to contain the identity element and be of primecardinality.

In this lattice tiling of the plane bycongruent squares, the green and violet

squares meet edge-to-edge as do the blueand orange squares.

An equivalent statement on homogeneous linear forms was originallyconjectured by Hermann Minkowski. A consequence is Minkowski'sconjecture on lattice tilings, which says that in any lattice tiling of space bycubes, there are two cubes that meet face to face. Keller's conjecture is thesame conjecture for non-lattice tilings, which turns out to be false in highdimensions.

References

• G. Hajós: Über einfache und mehrfache Bedeckung des'n'-dimensionalen Raumes mit einem Würfelgitter, Math. Z., 47(1941),427–467.

• H. Minkowski: Diophantische Approximationen, Leipzig, 1907.• L. Rédei, Die neue Theorie der endlichen abelschen Gruppen und

Verallgemeinerung des Hauptsatzes von Hajόs, Acta Math. Acad. Sci. Hung., 16 (1965), 329–373.• Stein, Sherman K. (1974), "Algebraic tiling" [1], The American Mathematical Monthly 81: 445–462,

ISSN 0002-9890, MR0340063• Stein, Sherman K.; Szabó, Sándor (1994), Algebra and tiling [2], Carus Mathematical Monographs, 25,

Mathematical Association of America, ISBN 978-0-88385-028-2, MR1311249

References[1] http:/ / www. jstor. org/ stable/ 2318582[2] http:/ / books. google. com/ books?id=QOa-mnX5Y4QC

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Harish-Chandra isomorphism 128

Harish-Chandra isomorphismIn mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism ofcommutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z(U(g)) of theuniversal enveloping algebra U(g) of a reductive Lie algebra g to the elements S(h)W of the symmetric algebra S(h)of a Cartan subalgebra h that are invariant under the Weyl group W.

Fundamental invariantsLet n be the rank of g, which is the dimension of the Cartan subalgebra h. H. S. M. Coxeter observed that S(h)W is apolynomial algebra in n variables (see Chevalley–Shephard–Todd theorem for a more general statement). Therefore,the center of the universal enveloping algebra of a reductive Lie algebra is a polynomial algebra. The degrees of thegenerators are the degrees of the fundamental invariants given in the following table.

Lie algebra Coxeter number h Dual Coxeter number Degrees of fundamental invariants

R 0 0 1

An

n + 1 n + 1 2, 3, 4, ..., n + 1

Bn

2n 2n − 1 2, 4, 6, ..., 2n

Cn

2n n + 1 2, 4, 6, ..., 2n

Dn

2n − 2 2n − 2 n; 2, 4, 6, ..., 2n − 2

E6

12 12 2, 5, 6, 8, 9, 12

E7

18 18 2, 6, 8, 10, 12, 14, 18

E8

30 30 2, 8, 12, 14, 18, 20, 24, 30

F4

12 9 2, 6, 8, 12

G2

6 4 2, 6

For example, the center of the universal enveloping algebra of G2 is a polynomial algebra on generators of degrees 2and 6.

Examples• If g is the Lie algebra sl2(R), then the center of the universal enveloping algebra is generated by the Casimir

invariant of degree 2, and the ring of invariants of the Weyl group is also generated by an element of degree 2.

Introduction and settingLet be a semisimple Lie algebra, its Cartan subalgebra and be two elements of the weight space andassume that a set of positive roots have been fixed. Let , resp. be highest weight modules with highestweight , resp. .

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Central characters

The -modules and are representations of the universal enveloping algebra and its center acts on themodules by scalar multiplication (this follows from the fact that the modules are generated by a highest weightvector). So, for and ,

and similarly for .The functions are homomorphims to scalars called central characters.

Statement of Harish-Chandra theoremFor any , the characters if and only if and are on the same orbit of the Weyl group of

under the affine action (corresponding to the choice of the positive roots ).Another closely related formulation is that the Harish-Chandra homomorphism from the centrum of the universalenveloping algebra to (invariant polynomials over the Cartan subalgebra fixed by the affineaction of the Weyl group) is an isomorphism.

ApplicationsThe theorem may be used to obtain a simple algebraic proof of Weyl's character formula for finite dimensionalrepresentations.Further, it is a necessary condition for the existence of a nonzero homomorphism of some highest weight moules (ahomomorphism of such modules preserves central character). A simple consequence is that for Verma modules orgeneralized Verma modules with highest weight , there exist only finitely many weights such that anonzero homomorphism exists.

References• Harish-Chandra (1951), "On some applications of the universal enveloping algebra of a semisimple Lie algebra",

Transactions of the American Mathematical Society 70: 28–96, ISSN 0002-9947, JSTOR 1990524, MR0044515• Humphreys, James E. (2000), Introduction to Lie algebras and representation theory, Birkhäuser, p. 126,

ISBN 978-0387900537• Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O, AMS, p. 26,

ISBN 978-0821846780• Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton

Mathematical Series, 45, Princeton University Press, ISBN 978-0-691-03756-1, MR1330919• Knapp, Anthony, Lie groups beyond an introduction, Second edition, pages 300–303.

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Hasse norm theorem 130

Hasse norm theoremIn number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if anonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be anelement k of K such that there is an element l of L with ; in other words k is a relative norm ofsome element of the extension field L. To be a local norm means that for some prime p of K and some prime P of Llying over K, then k is a norm from L

P; here the "prime" p can be an archimedean valuation, and the theorem is a

statement about completions in all valuations, archimedean and non-archimedean.The theorem is no longer true in general if the extension is abelian but not cyclic. A counter-example is given by thefield where every rational square is a local norm everywhere but is not a global norm.

This is an example of a theorem stating a local-global principle, and is due to Helmut Hasse.

References• H. Hasse, "A history of class field theory", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory,

Academic Press, 1973. Chap.XI.• G. Janusz, Algebraic number fields, Academic Press, 1973. Theorem V.4.5, p.156

Hasse–Arf theoremIn mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of afiltration of the Galois group of a finite Galois extension. A special case of it was originally proved by HelmutHasse,[1] [2] and the general result was proved by Cahit Arf.[3]

Statement

Higher ramification groupsThe theorem deals with the upper numbered higher ramification groups of a finite abelian extension L/K. So assumeL/K is a finite Galois extension, and that vK is a discrete normalised valuation of K, whose residue field hascharacteristic p > 0, and which admits a unique extension to L, say w. Denote by vL the associated normalisedvaluation ew of L and let be the valuation ring of L under vL. Let L/K have Galois group G and define the s-thramification group of L/K for any real s ≥ −1 by

So, for example, G−1 is the Galois group G. To pass to the upper numbering one has to define the function ψL/Kwhich in turn is the inverse of the function ηL/K defined by

The upper numbering of the ramification groups is then defined by Gt(L/K) = Gs(L/K) where s = ψL/K(t).These higher ramification groups Gt(L/K) are defined for any real t ≥ −1, but since vL is a discrete valuation, thegroups will change in discrete jumps and not continuously. Thus we say that t is a jump of the filtration{Gt(L/K) : t ≥ −1} if Gt(L/K) ≠ Gu(L/K) for any u > t. The Hasse–Arf theorem tells us the arithmetic nature of thesejumps

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HasseArf theorem 131

Statement of the theoremWith the above set up, the theorem states that the jumps of the filtration {Gt(L/K) : t ≥ −1} are all rational integers.

ExampleSuppose G is cyclic of order , residue characteristic and be the subgroup of of order . The thetheorem says that there exist positive integers such that

...[4]

Notes[1] H. Hasse, Führer, Diskriminante und Verzweigunsgskörper relativ Abelscher Zahlkörper, J. Reine Angew. Math. 162 (1930), pp.169–184.[2] H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci.

Tokyo 2 (1934), pp.477–498.[3] C. Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1940),

pp.1–44.[4] Serre, 4.3

References• Jürgen Neukirch, Algebraic Number Theory, Springer (1999).• (1980), Local Fields, Berlin, New York: Springer-Verlag, ISBN 9780387904245, MR0554237

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Hilbert's basis theoremIn mathematics, Hilbert's basis theorem, states that every ideal in the ring of multivariate polynomials over aNoetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic setover a field can be described as the set of common roots of finitely many polynomial equations. Hilbert (1890)proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finitegeneration of rings of invariants.Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give analgorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. Onecan determine basis polynomials using the method of Gröbner bases.

ProofThe following more general statement will be proved.

Theorem. If is a left- (respectively right-) Noetherian ring, then the polynomial ring is also a left-(respectively right-) Noetherian ring.It suffices to consider just the "Left" case.Proof (Theorem)

Suppose per contra that were a non-finitely generated left-ideal. Then it would be that by recursion(using the countable axiom of choice) that a sequence of polynomials could be found so that, letting

of minimal degree. It is clear that is a non-decreasingsequence of naturals. Now consider the left-ideal over where the are the leadingcoefficients of the . Since is left-Noetherian, we have that must be finitely generated; and since the comprise an -basis, it follows that for a finite amount of them, say will suffice. So for example,

some Now consider whose leadingterm is equal to that of moreover so of degree contradicting minimality. (Thm)

A constructive proof (not invoking the axiom of choice) also exists.Proof (Theorem):

Let be a left-ideal. Let be the set of leading coefficients of members of This is obviously aleft-ideal over and so is finitely generated by the leading coefficients of finitely many members of say

Let Let be the set of leading coefficients of members of whose degree

is As before, the are left-ideals over and so are finitely generated by the leading coefficients of finitelymany members of say with degrees Now let be the left-ideal generated

by We have and claim also Suppose per contra this were not so. Then let be of minimal degree, and denote its leading coefficientby Case 1: Regardless of this condition, we have so is a left-linear combination of the coefficients of the Consider which has the same leading term as

moreover so of degree contradicting minimality.

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Case 2: Then so is a left-linear combination of the leading

coefficients of the Considering we yield a similar contradiction as in

Case 1.Thus our claim holds, and which is finitely generated.

(Thm)

Note that the only reason we had to split into two cases was to ensure that the powers of multiplying the factors,were non-negative in the constructions.

ApplicationsLet be a Nötherian commutative ring. Hilbert's basis theorem has some immediate corollaries. First, byinduction we see that will also be Nötherian. Second, since any affine variety over (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal

and further as the locus of its generators, it follows that every affine variety is thelocus of finitely many polynomials — i.e. the interesection of finitely many hypersurfaces. Finally, if were afinitely-generated -algebra, then we know that (i.e. mod-ing out byrelations), where a set of polynomials. We can assume that is an ideal and thus is finitely generated. So would be a free -algebra (on generators) generated by finitely many relations

.

Mizar SystemThe Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in theHILBASIS file [1].

References• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.• Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen 36 (4): 473–534,

doi:10.1007/BF01208503, ISSN 0025-5831

References[1] http:/ / www. mizar. org/ JFM/ Vol12/ hilbasis. html

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Hilbert's irreducibility theoremIn number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number ofirreducible polynomials in a finite number of variables and having rational number coefficients admit a commonspecialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible.This theorem is a prominent theorem in number theory.

Formulation of the theoremHilbert's irreducibility theorem. Let

be irreducible polynomials in the ring

Then there exists an r-tuple of rational numbers (a1,...,ar) such that

are irreducible in the ring

Remarks.

• It follows from the theorem that there are infinitely many r-tuples. In fact the set of all irreducible specialization,called Hilbert set, is large in many senses. For example, this set is Zariski dense in

• There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if wedemand (a1,...,ar) to be integers.

• There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, global fieldsare Hilbertian.

• The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g.,it suffices to take in the definition. A recent result of Bary-Soroker shows that for a field K tobe Hilbertian it suffices to consider the case of and absolutely irreducible, that is,irreducible in the ring Kalg[X,Y], where Kalg is the algebraic closure of K.

ApplicationsHilbert's irreducibility theorem has numerous applications in number theory and algebra. For example:• The inverse Galois problem, Hilbert's original motivation. The theorem almost immediately implies that if a finite

group G can be realized as the Galois group of a Galois extension N of

then it can be specialized to a Galois extension N0 of the rational numbers with G as its Galois group. (To seethis, choose a monic irreducible polynomial f(X1,…,Xn,Y) whose root generates N over E. If f(a1,…,an,Y) isirreducible for some ai, then a root of it will generate the asserted N0.)

• Construction of elliptic curves with large rank.• Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's last theorem.

• If a polynomial is a perfect square for all large integer values of x, then g(x) is the square of apolynomial in . This follows from Hilbert's irreducibility theorem with and

.(More elementary proofs exist.) The same result is true when "square" is replaced by "cube", "fourth power", etc.

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GeneralizationsIt has been reformulated and generalized extensively, by using the language of algebraic geometry. See thin set(Serre).

References• J. P. Serre, Lectures on The Mordell-Weil Theorem, Vieweg, 1989.• M. D. Fried and M. Jarden, Field Arithmetic, Springer-Verlag, Berlin, 2005.• H. Völklein, Groups as Galois Groups, Cambridge University Press, 1996.• G. Malle and B. H. Matzat, Inverse Galois Theory, Springer, 1999.

Hilbert's NullstellensatzHilbert's Nullstellensatz (German: "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem whichmakes precise a fundamental relationship between the geometric and algebraic sides of algebraic geometry, animportant branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closedfields. The theorem was first proved by David Hilbert, after whom it is named.

FormulationLet k be a field (such as the rational numbers) and K be an algebraically closed field extension (such as the complexnumbers), consider the polynomial ring k[X1,X2,..., Xn] and let I be an ideal in this ring. The affine variety V(I)defined by this ideal consists of all n-tuples x = (x1,...,xn) in Kn such that f(x) = 0 for all f in I. Hilbert'sNullstellensatz states that if p is some polynomial in k[X1,X2,..., Xn] which vanishes on the variety V(I), i.e. p(x) = 0for all x in V(I), then there exists a natural number r such that pr is in I.An immediate corollary is the "weak Nullstellensatz": The ideal I in k[X1,X2,..., Xn] contains 1 if and only if thepolynomials in I do not have any common zeros in Kn.When k=K the "weak Nullstellensatz" may also be stated as follows: if I is a proper ideal in K[X1,X2,..., Xn], thenV(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal. This is the reason for thename of the theorem, which can be proved easily from the 'weak' form using the Rabinowitsch trick. The assumptionthat K be algebraically closed is essential here; the elements of the proper ideal (X2 + 1) in R[X] do not have acommon zero. With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as

for every ideal J. Here, denotes the radical of J and I(U) is the ideal of all polynomials which vanish on the setU.In this way, we obtain an order-reversing bijective correspondence between the affine varieties in Kn and the radicalideals of K[X1,X2,..., Xn]. In fact, more generally, one has a Galois connection between subsets of the space andsubsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators.

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GeneralizationThis generalization is due to Bourbaki, and is the most general form of the Nullstellensatz.Let be a Jacobson ring. If is a finitely generated R-algebra, then is a Jacobson ring. Further, if is amaximal ideal, then is a maximal ideal of R, and is a finite extension field of .Another generalization states that a faithfully flat morphism locally of finite type with X

quasi-compact has a quasi-section, i.e. there exists affine and faithfully flat and quasi-finite over X together withan X-morphism .

Applications

Commuting matricesThe fact that commuting matrices have a common eigenvector – and hence by induction stabilize a common flag andare simultaneously triangularizable – can be interpreted as a result of the weak Nullstellensatz, as follows:commuting matrices form a commutative algebra

over the matrices satisfy various polynomials such as their minimal polynomials, which form a proper ideal (because theyare not all zero, in which case the result is trivial); one might call this the characteristic ideal, by analogy with thecharacteristic polynomial.One then defines an eigenvector for a commutative algebra as a vector v such that for all one has

for a linear functional

This simply linearizes the definition of an eigenvalue, and is the correct definition for a common eigenvector, as if vis a common eigenvector, meaning then the functional is defined as

(treating scalars as multiples of the identity matrix , which has eigenvalue 1 for all vectors), andconversely an eigenvector for such a functional is a common eigenvector. Geometrically, the eigenvaluecorresponds to the point in affine k-space with coordinates with respect to the basis given by Then the existence of an eigenvalue is equivalent to the ideal generated by (the relations satisfied by) beingnon-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix overan algebraically closed field by showing that the characteristic polynomial has a zero.

References• Shigeru Mukai; William Oxbury (translator) (2003). An Introduction to Invariants and Moduli. Cambridge

studies in advanced mathematics. 81. p. 82. ISBN 0-521-80906-1.• David Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, New York : Springer-Verlag,

1999.

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Hilbert's syzygy theoremIn mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890)in connection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relationsbetween polynomial invariants, then relations between the relations, and so on, it explains how far one has to go toreach a clarified situation. It is now considered to be an early result of homological algebra, and through the depthconcept, to be a measure of the non-singularity of affine space.

Formal statementA contemporary formal statement is the following. Let k be a field and M a finitely generated module over thepolynomial ring

Hilbert's syzygy theorem then states that there exists a free resolution of M of length at most n.

References• David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics,

150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 MR1322960

Hilbert's Theorem 90In abstract algebra, Hilbert's Theorem 90 (or Satz 90) refers to an important result on cyclic extensions of fields (orto one of its generalizations) that leads to Kummer theory. In its most basic form, it tells us that if L/K is a cyclicextension of fields with Galois group G =Gal(L/K) generated by an element s and if a is an element of L of relativenorm 1, then there exists b in L such that

a = s(b)/b.The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's famous Zahlbericht of 1897,although it is originally due to Kummer. Often a more general theorem due to Emmy Noether is given the name,stating that if L/K is a finite Galois extension of fields with Galois group G =Gal(L/K), then the first cohomologygroup is trivial:

H1(G, L×) = {1}

ExamplesLet L/K be the quadratic extension . The Galois group is cyclic of order 2, its generator s is acting viaconjugation:

An element in L has norm . An element of norm one corresponds to a rationalsolution of the equation a2 +b2=1 or in other words, a point with rational coordinates on the unit circle. Hilbert'sTheorem 90 then states that every element y of norm one can be parametrized (with integral c,d) as

which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points on the unit circle correspond to Pythagorean triples, i.e. triples of

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integers satisfying .

CohomologyThe theorem can be stated in terms of group cohomology: if L× is the multiplicative group of any (not necessarilyfinite) Galois extension L of a field K with corresponding Galois group G, then

H1(G, L×) = {1}.A further generalization using non-abelian group cohomology states that if H is either the general or special lineargroup over L, then

H1(G,H) = {1}.This is a generalization since L× = GL1(L).

Another generalization is for X a scheme, and another one to MilnorK-theory plays a role in Voevodsky's proof of the Milnor conjecture.

References• Chapter II of J.S. Milne, Class Field Theory, available at his website [1].• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der

Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196

References[1] http:/ / www. jmilne. org/ math

Hopkins–Levitzki theoremIn the branch of abstract algebra called ring theory, the Akizuki-Hopkins–Levitzki theorem connects thedescending chain condition and ascending chain condition in modules over semiprimary rings. A ring R is calledsemiprimary if R/J(R) is semisimple and J(R) is a nilpotent ideal, where J(R) denotes the Jacobson radical. Thetheorem states that if R is a semiprimary ring and M is an R module, the three module conditions Noetherian,Artinian and "has a composition series" are equivalent. Without the semiprimary condition, the only true implicationis that if M has a composition series, then M is both Noetherian and Artinian.The theorem takes its current form from a paper by Charles Hopkins and a paper by Jacob Levitzki, both in 1939.For this reason it is often cited as the Hopkins–Levitzki theorem. However Yasuo Akizuki is sometimes includedsince he proved the result for commutative rings a few years earlier (Lam 2001).Since it is known that right Artinian rings are semiprimary, a direct corollary of the theorem is: a right Artinian ringis also right Noetherian. The analogous statement for left Artinian rings holds as well. This is not true in general forArtinian modules, because there are examples of Artinian modules which are not Noetherian.Another direct corollary is that if R is right Artinian, then R is left Artinian if and only if it is left Noetherian.

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References• Charles Hopkins (1939) Rings with minimal condition for left ideals, Ann. of Math. (2) 40, pages 712–730.• T. Y. Lam (2001) A first course in noncommutative rings, Springer-Verlag. page 55 ISBN 0-387-95183-0• Jakob Levitzki (1939) On rings which satisfy the minimum condition for the right-hand ideals, Compositio Math.

7, pages 214–222.

Hurwitz's theorem (normed division algebras)In algebra, Hurwitz's theorem (also called the “1,2,4 8 Theorem”), named after Adolf Hurwitz, who proved it in1898, states: Every normed division algebra with an identity is isomorphic to one of the following four algebras: R,C, H and O, that is the real numbers, the complex numbers, the quaternions and the octonions.[1] [2] Theclassification of real division algebras began with Georg Frobenius,[3] continued with Hurwitz[4] and was set ingeneral form by Max Zorn.[5] A brief historical summary may be found in Badger.[6]

A full proof can be found in Kantor and Solodovnikov,[7] and in Shapiro.[8] As a basic idea, if an algebra A isproportional to 1 then it is isomorphic to the real numbers. Otherwise we extend the subalgebra isomorphic to 1using the Cayley–Dickson construction and introducing a vector e which is orthogonal to 1. This subalgebra isisomorphic to the complex numbers. If this is not all of A then we once again use the Cayley–Dickson constructionand another vector orthogonal to the complex numbers and get a subalgebra isomorphic to the quaternions. If this isnot all of A then we double up once again and get a subalgebra isomorphic to the Cayley numbers (or Octonions).We now have a theorem which says that every subalgebra of A that contains 1 and is not A is associative. The Cayleynumbers are not associative and therefore must be A.Hurwitz's theorem can be used to prove that the product of the sum of n squares by the sum of n squares is the sumof n squares in a bilinear way only when n is equal to 1, 2, 4 and 8.[9]

In-line references[1] JA Nieto and LN Alejo-Armenta (2000). "Hurwitz theorem and parallelizable spheres from tensor analysis". Arxiv preprint hep-th/0005184.

arXiv:hep-th/0005184.[2] Kevin McCrimmon (2004). "Hurwitz's theorem 2.6.2" (http:/ / books. google. com/ books?id=6YG4ycpKMYkC& pg=PA166). A taste of

Jordan algebras. Springer. p. 166. ISBN 0387954473. . "Only recently was it established that the only finite-dimensional real nonassociativedivision algebras have dimensions 1,2,4,8; the algebras were not classified, and the proof was topological rather than algebraic."

[3] Georg Frobenius (1878). "Über lineare Substitutionen und bilineare Formen". J. Reine Angew. Math. 84: 1–63.[4] Hurwitz, A. (1898). "Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln (On the composition of quadratic forms

of arbitrary many variables)" (in German). Nachr. Ges. Wiss. Göttingen: 309–316. JFM 29.0177.01.[5] Max Zorn (1930). "Theorie der alternativen Ringe". Abh. Math. Sem. Univ. Hamburg 8: 123–147.[6] Matthew Badger. "Division algebras over the real numbers" (http:/ / www. math. washington. edu/ ~mbadger/ divalg3. pdf). .[7] IL Kantor and AS Solodovnikov (1989). "Normed algebras with an identity. Hurwitz's theorem." (http:/ / books. google. com/ books?as_q=&

num=10& btnG=Google+ Search& as_epq=Normed+ algebras+ with+ an+ identity. + Hurwitz's+ theorem& as_oq=& as_eq=& as_brr=0&as_pt=ALLTYPES& lr=& as_vt=& as_auth=& as_pub=& as_sub=& as_drrb_is=q& as_minm_is=0& as_miny_is=& as_maxm_is=0&as_maxy_is=& as_isbn=& as_issn=). Hypercomplex numbers. An elementary introduction to algebras (2nd ed.). Springer-Verlag. p. 121.ISBN 0387969802. .

[8] Daniel B. Shapiro (2000). "Appendix to Chapter 1. Composition algebras" (http:/ / books. google. com/ books?id=qrFhUda9JbkC&pg=PA21). Compositions of quadratic forms. Walter de Gruyter. pp. 21 ff. ISBN 311012629X. .

[9] Joe Roberts (1992). "Square identities" (http:/ / books. google. com/ books?id=DvX90EKMxGwC& pg=PA93). Lure of the integers.Cambridge University Press. ISBN 088385502X. .

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Background references• John H. Conway, Derek A. Smith On Quaternions and Octonions. A.K. Peters, 2003.• John Baez, The Octonions (http:/ / math. ucr. edu/ home/ baez/ octonions/ ), AMS 2001.

Isomorphism extension theoremIn field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding theextension of a field isomorphism to a larger field.

Isomorphism extension theoremThe theorem states that given any field , an algebraic extension field of and an isomorphism mapping

onto a field then can be extended to an isomorphism mapping onto an algebraic extension of(a subfield of the algebraic closure of ).

The proof of the isomorphism extension theorem depends on Zorn's lemma.

References• D.J. Lewis, Introduction to algebra, Harper & Row, 1965, Chap.IV.12, p.193.

Isomorphism theoremIn mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe therelationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings,vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphismtheorems can be generalized to the context of algebras and congruences.

HistoryThe isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noetherin her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern which was publishedin 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of RichardDedekind and previous papers by Noether.Three years later, B.L. van der Waerden published his influential Algebra, the first abstract algebra textbook thattook the now-traditional groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noetheron group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, OttoSchreier, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, calledhomomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.

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GroupsWe first state the three isomorphism theorems in the context of groups. Note that some sources switch the numberingof the second and third theorems.[1] Sometimes, the lattice theorem is referred to as the fourth isomorphism theoremor the correspondence theorem.

Statement of the theorems

First isomorphism theorem

Let G and H be groups, and let φ: G → H be a homomorphism. Then:1. The kernel of φ is a normal subgroup of G,2. The image of φ is a subgroup of H, and3. The image of φ is isomorphic to the quotient group G / ker(φ).In particular, if φ is surjective then H is isomorphic to G / ker(φ).

Second isomorphism theorem

Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then:1. The product SN is a subgroup of G,2. The intersection S ∩ N is a normal subgroup of S, and3. The quotient groups (SN) / N and S / (S ∩ N) are isomorphic.Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the normalizer of N. Inthis case, the intersection S ∩ N is not a normal subgroup of G, but it is still a normal subgroup of S.

Third isomorphism theorem

Let G be a group. Let N and K be normal subgroups of G, withK ⊆ N ⊆ G.

Then1. The quotient N / K is a normal subgroup of the quotient G / K, and2. The quotient group (G / K) / (N / K) is isomorphic to G / N.

Discussion

First isomorphism theorem

The first isomorphism theorem follows from the category theoretical fact that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism f: G→H. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into , where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is

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represented in the diagram by an object and a monomorphism (kernels are always monomorphisms), which completethe short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequenceconvention saves us from having to draw the zero morphisms from to H and .If the sequence is right split (i. e., there is a morphism σ that maps to a π-preimage of itself), then G isthe semidirect product of the normal subgroup and the subgroup . If it is left split (i. e., there existssome such that ), then it must also be right split, and is a directproduct decomposition of G. In general, the existence of a right split does not imply the existence of a left split; butin an abelian category (such as the abelian groups), left splits and right splits are equivalent by the splitting lemma,and a right split is sufficient to produce a direct sum decomposition . In an abelian category, allmonomorphisms are also normal, and the diagram may be extended by a second short exact sequence

.In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while theintersection S ∩ N is the meet.The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general mapsbetween objects. It is sometimes informally called the "freshman theorem", because "even a freshman could figure itout: just cancel out the Ks!"

RingsThe statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion ofan ideal.

First isomorphism theoremLet R and S be rings, and let φ: R → S be a ring homomorphism. Then:1. The kernel of φ is an ideal of R,2. The image of φ is a subring of S, and3. The image of φ is isomorphic to the quotient ring R / ker(φ).In particular, if φ is surjective then S is isomorphic to R / ker(φ).

Second isomorphism theoremLet R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:1. The sum S + I = {s + i | s ∈ S, i ∈ I} is a subring of R,2. The intersection S ∩ I is an ideal of S, and3. The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic.

Third isomorphism theoremLet R be a ring. Let A and B be ideals of R, with

B ⊆ A ⊆ R.Then1. The set A / B is an ideal of the quotient R / B, and2. The quotient ring (R / B) / (A / B) is isomorphic to R / A.

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ModulesThe statements of the isomorphism theorems for modules are particularly simple, since it is possible to form aquotient module from any submodule. The isomorphism theorems for vector spaces and abelian groups are specialcases of these. For vector spaces, all of these theorems follow from the rank-nullity theorem.For all of the following theorems, the word “module” will mean “R-module”, where R is some fixed ring.

First isomorphism theoremLet M and N be modules, and let φ: M → N be a homomorphism. Then:1. The kernel of φ is a submodule of M,2. The image of φ is a submodule of N, and3. The image of φ is isomorphic to the quotient module M / ker(φ).In particular, if φ is surjective then N is isomorphic to M / ker(φ).

Second isomorphism theoremLet M be a module, and let S and T be submodules of M. Then:1. The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M,2. The intersection S ∩ T is a submodule of S, and3. The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.

Third isomorphism theoremLet M be a module. Let S and T be submodules of M, with

T ⊆ S ⊆ M.Then1. The quotient S / T is a submodule of the quotient M / T, and2. The quotient (M / T) / (S / T) is isomorphic to M / S.

GeneralTo generalise this to universal algebra, normal subgroups need to be replaced by congruences.Briefly, if is an algebra, a congruence on is an equivalence relation on which is a subalgebra whenconsidered as a subset of (the latter with the coordinate-wise operation structure). One can make the set ofequivalence classes into an algebra of the same type by defining the operations via representatives; this will bewell-defined since is a subalgebra of .

First Isomorphism Theorem

If and are algebras, and is a homomorphism , then the equivalence relation on defined by if and only if is a congruence on , and the algebra is isomorphic to theimage of , which is a subalgebra of .

Second Isomorphism Theorem

Given an algebra , a subalgebra of , and a congruence on , we let be the subset of determined by all congruence classes that contain an element of , and we let be the intersection of (considered as a subset of ) with . Then is a subalgebra of , is a congruence on

, and the algebra is isomorphic to the algebra .

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Third Isomorphism TheoremLet be an algebra, and let and be two congruence relations on , with contained in . Then determines a congruence on defined by if and only if and are equivalent modulo (where represents the -equivalence class of ), and is isomorphic to .

Notes[1] Jacobson (2009), p. 101, use "first" for the isomorphism of the modules (S + T) / T and S / (S ∩ T), and "second" for (M / T) / (S / T) and

M / S.

References• Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern,

Mathematische Annalen 96 (1927) p. 26-61• Colin McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the rise of functors' in The

Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and JoséFerreirós), Oxford University Press (2006) p. 211–35.

• Jacobson, Nathan (2009), Basic algebra, 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7

External links• First isomorphism theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=1114) on

PlanetMath. Proof of first isomorphism theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects&amp;id=2922) on PlanetMath

• Second isomorphism theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=1334) onPlanetMath. Proof of second isomorphism theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects&amp;id=3153) on PlanetMath

• Third isomorphism theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=1126) onPlanetMath. Proof of third isomorphism theorem (http:/ / planetmath. org/ ?op=getobj& amp;from=objects&amp;id=7496) on PlanetMath

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Jacobson density theoremIn mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobsondensity theorem is a theorem concerning simple modules over a ring R.[1]

The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of lineartransformations of a vector space[2] [3] . This theorem first appeared in the literature in 1945, in the famous paper"Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.[4] This can be viewed as akind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings.

Motivation and formal statementLet R be a ring and let U be a simple right R-module. If u is a non-zero element of U, u·R = U (where u·R is thecyclic submodule of U generated by u). Therefore, if u and v are non-zero elements of U, there is an element of Rthat induces an endomorphism of U transforming u to v. The natural question now is whether this can be generalizedto arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple (x1, ...,xn) and (y1, ..., yn) separately, so that there is an element of R with the property that xi·r = yi for all i. If D is the set ofall R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson densitytheorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.With the above in mind, theorem may be stated this way:The Jacobson Density Theorem

Let U be a simple right R-module and write D = EndR(U). Let A be any D-linear transformation on U and letX be a finite D-linearly independent subset of U. Then there exists an element r of R such that A(x) = x·r for allx in X.[5]

ProofIn the Jacobson density theorem, the right R-module U is simultaneously viewed as a left D-module whereD=EndR(U) module in the natural way: the action g·u is defined to be g(u). It can be verified that this is indeed a leftmodule structure on U.[6] As noted before, Schur's lemma proves D is a division ring if U is simple, and so U is avector space over D.The proof also relies on the following theorem proven in (Isaacs 1993) p. 185:Theorem

Let U be a simple right R-module and let D = EndR(U) - the set of all R module endomorphisms of U. Let X bea finite subset of U and write I = annR(X) - the annihilator of X in R. Let u be in U with u·I = 0. Then u is inXD; the D-span of X.

Proof (of the Jacobson density theorem)We proceed by mathematical induction on the number n of elements in X. If n=0 so that X is empty, then thetheorem is vacuously true and the base case for induction is verified. Now we assume that X is non-empty withcardinality n. Let x be an element of X and write Y = X \ {x}. If A is any D-linear transformation on U, theinduction hypothesis guarantees that there exists an s in R such that A(y) = y·s for all y in Y.Write I = annR(Y). It is easily seen that x·I is a submodule of U. If it were the case that x·I = 0, then theprevious theorem would indicate that x would be in the D-span of Y. This would contradict the linearindependence of X, so it must be that x·I ≠ 0. So, by simplicity of U, the submodule x·I = U. Since A(x) - x·s isin U=x·I, there exists i in I such that x·i = A(x) - x·s.

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After defining r = s + i, we compute that y·r = y·(s + i) = y·s + y·i = y·s = A(y) for all y in Y.[7] Also, x·r = x·(s +i) = x·s + A(x) - x·s = A(x). Therefore, A(z) = z·r for all z in X, as desired. This completes the inductive step ofthe proof. It follows now from mathematical induction that the theorem is true for finite sets X of any size.

Topological characterizationA ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson densitytheorem.[8] There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring ofEnd(DU) by identifying each element of R with the D linear transformation it induces by right multiplication. If U isgiven the discrete topology, and if UU is given the product topology, and End(DU) is viewed as a subspace of UU andis given the subspace topology, then R acts densely on U if and only if R is dense set in End(DU) with thistopology[9] .

ConsequencesThe Jacobson density theorem has various important consequences in the structure theory of rings.[10] Notably, theArtin–Wedderburn theorem's conclusion about the structure of simple right Artinian rings is recovered. TheJacobson density theorem also characterizes right or left primitive rings as dense subrings of the ring of D-lineartransformations on some D- vector space U, where D is a division ring.[11]

Relations to other resultsThis result is related to the Von Neumann bicommutant theorem, which states that, for a *-algebra A of operators ona Hilbert space H, the double commutant A′′ can be approximated by A on any given finite set of vectors. See alsothe Kaplansky density theorem in the von Neumann algebra setting.

Notes[1] Isaacs, p. 184[2] Such rings of linear transformations are also known as full linear rings.[3] Isaacs, Corollary 13.16, p. 187[4] Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions" (http:/ / www. jstor. org/ pss/ 1990204)[5] Isaacs, Theorem 13.14, p. 185[6] Incidentally it is also a D-R bimodule structure.[7] Of course, y·i=0 by definition of I.[8] Herstein, Definition, p. 40[9] It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description.[10] Herstein, p. 41[11] Isaacs, Corollary 13.16, p. 187

References• I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America.

ISBN 0-88385-015-X.• I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company.

ISBN 0-534-19002-2.• Jacobson, N. (1945), "Structure theory of simple rings without finiteness assumptions", Trans. Amer. Math. Soc.

57: 228–245, ISSN 0002-9947, MR0011680

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External links• PlanetMath page (http:/ / planetmath. org/ encyclopedia/ JacobsonDensityTheorem. html)

Jordan's theorem (symmetric group)Jordan's theorem is a statement in finite group theory. It states that if a primitive permutation group G⊆Sn containsa p-cycle for a prime p<n-2, then G is either the whole symmetric group Sn or the alternating group An.

The statement can be generalized for p being a prime power.

References• Griess, Robert L. (1998), Twelve sporadic groups, Springer, p. 5, ISBN 978-3540627784• Isaacs, I. Martin (2008), Finite group theory, AMS, p. 245, ISBN 978-0821843444• Neuman, Peter M. (1975), "Primitive permutation groups containing a cycle of prime power length" [1], Bulletin

London Mathematical Society 7 (3): 298–299

External links• Jordan's Symmetric Group Theorem [2], mathworld

References[1] http:/ / blms. oxfordjournals. org/ content/ 7/ 3/ 298. extract[2] http:/ / mathworld. wolfram. com/ JordansSymmetricGroupTheorem. html

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JordanSchur theorem 148

Jordan–Schur theoremIn mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theoremin its original form due to Camille Jordan. In that form, it states that there is a function ƒ(n) such that given a finitegroup G that is a subgroup of the group of n-by-n complex matrices, then there is a subgroup H of G such that H isabelian, H is normal with respect to G and H has index at most ƒ(n). Schur proved a more general result that applieswhen G is assumed not to be finite but just periodic. Schur showed that ƒ(n) may be taken to be

((8n)1/2 + 1)2n2 − ((8n)1/2 − 1)2n2.[1]

A tighter bound (for n ≥ 3) is due to Speiser who showed that as long as G is finite, one can takeƒ(n) = n!12n(π(n+1)+1)

where π(n) is the prime-counting function.[1] [2] This was subsequently improved by Blitchfeldt who replaced the"12" with a "6". Unpublished work on the finite case was also done by Boris Weisfeiler.[3] Subsequently, MichaelCollins using the classification of finite simple groups showed that in the finite case, one can take f(n)) = (n+1)!when n is at least 71, and gave near complete descriptions of the behavior for smaller n.

References[1] Curtis, Charles; Reiner, Irving (1962). Representation Theory of Finite Groups and Associated Algebras. John Wiley & Sons. pp. 258–262.[2] Speiser, Andreas (1945). Die Theorie der Gruppen von endlicher Ordnung, mit Andwendungen auf algebraische Zahlen und Gleichungen

sowie auf die Krystallographie, von Andreas Speiser. New York: Dover Publications. pp. 216–220.[3] Collins, Michael J. (2007). "On Jordan’s theorem for complex linear groups". Journal of Group Theory 10 (4): 411–423.

doi:10.1515/JGT.2007.032.

• Ben Green lecture notes Analytic Topics in Group Theory, chapter 2 (http:/ / www. dpmms. cam. ac. uk/ ~bjg23/ATG. html)http:/ / www. dpmms. cam. ac. uk/ ~bjg23/ ATG/ Chapter2. pdf

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Krull's principal ideal theoremIn commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a boundon the height of a principal ideal in a Noetherian ring. The theorem is sometimes referred to by its German name,Krulls Hauptidealsatz.Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.This theorem can be generalized to ideals that are not principal, and the result is often called Krull's heighttheorem. This says that if R is a Noetherian ring and I is a proper ideal generated by n elements of R, then I hasheight at most n.

References• Matsumura, Hideyuki (1970), Commutative algebra, New York: Benjamin, see in particular section (12.I), p. 77

Krull–Schmidt theoremIn mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chainsof subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

DefinitionsWe say that a group G satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups ofG:

is eventually constant, i.e., there exists N such that GN = GN+1 = GN+2 = ... . We say that G satisfies the ACC onnormal subgroups if every such sequence of normal subgroups of G eventually becomes constant.Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasingsequences of (normal) subgroups:

Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group satisfies ACC butnot DCC, since (2) > (2)2 > (2)3 > ... is an infinite decreasing sequence of subgroups. On the other hand, the -torsion part of (the quasicyclic p-group) satisfies DCC but not ACC.We say a group G is indecomposable if it cannot be written as a direct product of non-trivial subgroups G = H × K.

Krull–Schmidt theoremThe theorem says:If is a group that satisfies ACC and DCC on normal subgroups, then there is a unique way of writing as adirect product of finitely many indecomposable subgroups of . Here, uniquenessmeans direct decompositions into indecomposable subgroups have the exchange property. That is: suppose

is another expression of as a product of indecomposable subgroups. Thenand there is a reindexing of the 's satisfying

• and are isomorphic for each ;• for each .

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Krull–Schmidt theorem for modulesIf is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or– equivalently – of finite length), then is a direct sum of indecomposable modules. Up to a permutation, theindecomposable components in such a direct sum are uniquely determined up to isomorphism.[1]

In general, the theorem fails, if one only assumes that the module is Noetherian.

HistoryThe present-day Krull–Schmidt theorem was first proved by Joseph Wedderburn (Ann. of Math (1909)), for finitegroups, though he mentions some credit is due to an earlier study of G.A. Miller where direct products of abeliangroups were considered. Wedderburn's theorem is stated as an exchange property between direct decompositions ofmaximum length. However, Wedderburn's proof makes no use of automorphisms.The thesis of Robert Remak (1911) derived the same uniqueness result as Wedderburn but also proved (in modernterminology) that the group of central automorphisms acts transitively on the set of direct decompositions ofmaximum length of a finite group. From that stronger theorem Remak also proved various corollaries including thatgroups with a trivial center and perfect groups have a unique Remak decomposition.Otto Schmidt (Sur les produits directs, S. M. F. Bull. 41 (1913), 161–164), simplified the main theorems of Remakto the 3 page predecessor to today's textbook proofs. His method improves Remak's use of idempotents to create theappropriate central automorphisms. Both Remak and Schmidt published subsequent proofs and corollaries to theirtheorems.Wolfgang Krull (Über verallgemeinerte endliche Abelsche Gruppen, M. Z. 23 (1925) 161–196), returned to G.A.Miller's original problem of direct products of abelian groups by extending to abelian operator groups with ascendingand descending chain conditions. This is most often stated in the language of modules. His proof observes that theidempotents used in the proofs of Remak and Schmidt can be restricted to module homomorphisms; the remainingdetails of the proof are largely unchanged.O. Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebrasby proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chainconditions. This proof makes no use of idempotents and does not reprove the transitivity of Remak's theorems.Kurosh's The Theory of Groups and Zassenhaus' The Theory of Groups include the proofs of Schmidt and Ore underthe name of Remak–Schmidt but acknowledge Wedderburn and Ore. Later texts use the title Krull–Schmidt(Hungerford's Algebra) and Krull–Schmidt–Azumaya (Curtis–Reiner). The name Krull–Schmidt is now popularlysubstituted for any theorem concerning uniqueness of direct products of maximum size. Some authors choose to calldirect decompositions of maximum-size Remak decompositions to honor his contributions.

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Notes[1] Jacobson (2009), p. 115.

References• Jacobson, Nathan (2009), Basic algebra, 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7

Further reading• Hungerford, Thomas W. Algebra, Graduate Texts in Mathematics Volume 73. ISBN 0-387-90518-9• A. Facchini: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules.

Progress in Mathematics, 167. Birkhäuser Verlag, Basel, 1998. ISBN 3-7643-5908-0• A. Facchini, D. Herbera, L.S. Levy, P. Vámos: Krull–Schmidt fails for Artinian modules. Proc. Amer. Math. Soc.

123 (1995), no. 12, 3587–3592.• C.M. Ringel: Krull–Remak–Schmidt fails for Artinian modules over local rings. Algebr. Represent. Theory 4

(2001), no. 1, 77–86.

External links• Page at PlanetMath (http:/ / planetmath. org/ encyclopedia/ KrullRemakSchmidtTheorem. html)

Künneth theoremIn mathematics, especially in homological algebra and algebraic topology, a Künneth theorem is a statementrelating the homology of two objects to the homology of their product. The classical statement of the Künneththeorem relates the singular homology of two topological spaces X and Y and their product space X × Y. In thesimplest possible case the relationship is that of a tensor product, but for applications it is very often necessary toapply certain tools of homological algebra to express the answer.A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the namehas become generic. These many results are named for the German mathematician Hermann Künneth.

Singular homology with coefficients in a fieldLet X and Y be two topological spaces. In general one uses singular homology; but if X and Y happen to be CWcomplexes, then this can be replaced by cellular homology, because that is isomorphic to singular homology. Thesimplest case is when the coefficient ring for homology is a field F. In this situation, the Künneth theorem (forsingular homology) states that for any integer k,

Furthermore, the isomorphism is a natural isomorphism. The map from the sum to the homology group of theproduct is called the cross product. More precisely, there is a cross product operation by which an i-cycle on X and aj-cycle on Y can be combined to create an (i+j)-cycle on X × Y; so that there is an explicit linear mapping definedfrom the direct sum to Hk(X × Y).A consequence of this result is that the Betti numbers, the dimensions of the homology with Q coefficients, of X × Ycan be determined from those of X and Y. If pZ(t) is the generating function of the sequence of Betti numbers bk(Z) ofa space Z, then

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Here when there are finitely many Betti numbers of X and Y, each of which is a natural number rather than ∞, thisreads as an identity on Poincaré polynomials. In the general case these are formal power series with possibly infinitecoefficients, and have to be interpreted accordingly. Furthermore, the above statement holds not only for the Bettinumbers but also for the generating functions of the dimensions of the homology over any field. (If the integerhomology is not torsion-free then these numbers may differ from the standard Betti numbers.)

Singular homology with coefficients in a PIDThe above formula is simple because vector spaces over a field have very restricted behavior. As the coefficient ringbecomes more general, the relationship becomes more complicated. The next simplest case is the case when thecoefficient ring is a principal ideal domain. This case is particularly important because the integers are a PID.In this case the equation above is no longer always true. A correction factor appears to account for the possibility oftorsion phenomena. This correction factor is expressed in terms of the Tor functor, the first derived functor of thetensor product.When R is a PID, then the correct statement of the Künneth theorem is that for any topological spaces X and Y thereare natural short exact sequences

Furthermore these sequences split, but not canonically.

ExampleThe short exact sequences just described can easily be used to compute the homology groups

with integer coefficients of the product RP2 x RP2 of two real projective planes. These spaces are CW complexes.Denoting the homology group by h

i for brevity's sake, one knows from a simple calculation with

cellular homology that

and hi

is zero for all other values of i. The only non-zero Tor group (torsion product) which can be formed fromthese values of h

i is

Therefore the Künneth short exact sequence reduces in every degree to an isomorphism, because there is a zerogroup in each case on either the left or the right side in the sequence. The result is

and all the other homology groups are zero.

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The Künneth spectral sequenceFor a general commutative ring R, the homology of X and Y is related to the homology of their product by a Künnethspectral sequence

In the cases described above, this spectral sequence collapses to give an isomorphism or a short exact sequence.

Relation with homological algebra, and idea of proofThe chain complex of the space X × Y is related to the chain complexes of X and Y by a natural quasi-isomorphism

For singular chains this is the theorem of Eilenberg and Zilber. For cellular chains on CW complexes, it is astraightforward isomorphism. Then the homology of the tensor product on the right is given by the spectral Künnethformula of homological algebra.[1]

The freeness of the chain modules means that in this geometric case it is not necessary to use any hyperhomology ortotal derived tensor product.There are analogues of the above statements for singular cohomology and sheaf cohomology. For sheaf cohomologyon an algebraic variety, Grothendieck found six spectral sequences relating the possible hyperhomology groups oftwo chain complexes of sheaves and the hyperhomology groups of their tensor product.[2]

Künneth theorems in generalized homology and cohomology theoriesThere are many generalized or extraordinary homology and cohomology theories for topological spaces. K-theoryand cobordism are the best-known. Their striking common feature (not their definition) is that they do not arise fromordinary chain complexes. Thus Künneth theorems can not be obtained by the above methods of homologicalalgebra. Nevertheless Künneth theorems in just the same form have been proved in very many cases by various othermethods. The first were Atiyah's Künneth theorem for complex K-theory and Conner and Floyd's result incobordism.[3] [4] A general method of proof emerged, based upon a homotopical theory of modules over highlystructured ring spectra.[5] The homotopy category of such modules closely resembles the derived categories ofhomological algebra.

References[1] See final chapter of Mac Lane, Saunders (1963), Homology, Berlin: Springer, ISBN 038703823X[2] Grothendieck, Alexander; Dieudonné, Jean (1963), "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) :

III. Étude cohomologique des faisceaux cohérents, Seconde partie" (http:/ / www. numdam. org:80/ numdam-bin/feuilleter?id=PMIHES_1963__17_), Publications Mathématiques de l'IHÉS 17: 5–91, (EGA III2, Théorème 6.7.3.).

[3] Atiyah, Michael F. (1967), K-theory, New York: W. A. Benjamin.[4] Conner, P. E.; Floyd, E. E. (1964), Differentiable periodic maps, Berlin: Springer.[5] Elmendorf, A. D.; Kriz, I.; Mandell, M. A. & May, J. P. (1997), Rings, modules and algebras in stable homotopy theory, Providence, RI:

American Mathematical Society, ISBN 0821806386.

External links• Hazewinkel, Michiel, ed. (2001), "Künneth formula" (http:/ / eom. springer. de/ k/ k056010. htm), Encyclopaedia

of Mathematics, Springer, ISBN 978-1556080104

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Kurosh subgroup theoremIn the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure ofsubgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russian mathematician, in1934.[1] Informally, the theorem says that every subgroup of a free product is itself a free product of a free group andof groups conjugate to the subgroups of the factors of the original free product.

History and generalizationsAfter the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem,including proofs of Kuhn (1952),[2] Mac Lane (1958)[3] and others. The theorem was also generalized for describingsubgroups of amalgamated free products and HNN extensions.[4] [5] Other generalizations include consideringsubgroups of free pro-finite products[6] and a version of the Kurosh subgroup theorem for topological groups.[7]

In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural results ofBass-Serre theory about groups acting on trees.[8]

Statement of the theoremLet G = A∗B be the free product of groups A and B and let H ≤ G be a subgroup of G. Then there exist a family (Ai)i

∈ I of subgroups Ai ≤ A, a family (Bj)j ∈ J of subgroups Bj ≤ B, families gi, i ∈ I and fj, j ∈ J of elements of G, and asubset X ⊆ G such that

This means that X freely generates a subgroup of G isomorphic to the free group F(X) with free basis X and that,moreover, giAigi

−1, fjBjfj−1 and X generate H in G as a free product of the above form.

There is a generalization of this to the case of free products with arbitrarily many factors.[9] Its formulation is:If H is a subgroup of ∗i∈IGi = G, then

where X ⊆ G and J is some index set and gj ∈ G and each Hj is a subgroup of some Gi.

Proof using Bass-Serre theoryThe Kurosh subgroup theorem easily follows from the basic structural results in Bass–Serre theory, as explained, forexample in the book of Cohen (1987)[8] :Let G = A∗B and consider G as the fundamental group of a graph of groups Y consisting of a single non-loop edgewith the vertex groups A and B and with the trivial edge group. Let X be the Bass-Serre universal covering tree forthe graph of groups Y. Since H ≤ G also acts on X, consider the quotient graph of groups Z for the action of H on X.The vertex groups of Z are subgroups of G-stabilizers of vertices of X, that is, they are conjugate in G to subgroupsof A and B. The edge groups of Z are trivial since the G-stabilizers of edges of X were trivial. By the fundamentaltheorem of Bass–Serre theory, H is canonically isomorphic to the fundamental group of the graph of groups Z. Sincethe edge groups of Z are trivial, it follows that H is equal to the free product of the vertex groups of Z and the freegroup F(X) which is the fundamental group (in the standard topological sense) of the underlying graph Z of Z. Thisimplies the conclusion of the Kurosh subgroup theorem.

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Notes[1] A. G. Kurosh, Die Untergruppen der freien Produkte von beliebigen Gruppen. Mathematische Annalen, vol. 109 (1934), pp. 647-660.[2] H. W. Kuhn. Subgroup theorems for groups presented by generators and relations. Annals of Mathematics (2), vol. 56, (1952), pp. 22-46[3] S. Mac Lane. A proof of the subgroup theorem for free products. Mathematika, vol. 5 (1958), pp. 13-19[4] A. Karrass, and D. Solitar. The subgroups of a free product of two groups with an amalgamated subgroup. Transactions of the American

Mathematical Society, vol. 150 (1970), pp. 227-255.[5] A. Karrass, and D. Solitar. Subgroups of HNN groups and groups with one defining relation. Canadian Journal of Mathematics, vol. 23

(1971), pp. 627-643.[6] Zalesskii, Pavel Aleksandrovich (1990). "[Open subgroups of free profinite products over a profinite space of indices]" (in Russian). Doklady

Akademii Nauk SSSR 34 (1): 17–20.[7] P. Nickolas. A Kurosh subgroup theorem for topological groups. Proceedings of the London Mathematical Society (3), vol. 42 (1981), no. 3,

pp. 461-477[8] Daniel Cohen. Combinatorial group theory: a topological approach. London Mathematical Society Student Texts, 14. Cambridge University

Press, Cambridge, 1989. ISBN 0-521-34133-7; 0-521-34936-2[9] William S. Massey, Algebraic topology: an introduction. (http:/ / books. google. com/ books?id=IX0dhDDHezgC& pg=PA218&

dq="Kurosh+ subgroup+ theorem"& as_brr=3& ei=dQ10S8zsKKasNaSNgJsE& cd=1#v=onepage& q="Kurosh subgroup theorem"& f=false)Graduate Texts in Mathematics, Springer-Verlag, New York, 1977, ISBN 0387902716; pp. 218–225

Lagrange's theorem (group theory)Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number ofelements) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.

Proof of Lagrange's TheoremThis can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certainequivalence relation on G and therefore form a partition of G. Specifically, x and y in G are related if and only ifthere exists h in H such that x = yh. If we can show that all cosets of H have the same number of elements, then eachcoset of H has precisely |H| elements. We are then done since the order of H times the number of cosets is equal tothe number of elements in G, thereby proving that the order of H divides the order of G. Now, if aH and bH are twoleft cosets of H, we can define a map f : aH → bH by setting f(x) = ba-1x. This map is bijective because its inverse isgiven by This proof also shows that the quotient of the orders |G| / |H| is equal to the index [G : H] (the number of left cosetsof H in G). If we write this statement as

then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice.

Using the theoremA consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integernumber k with ak = e, where e is the identity element of the group) divides the order of that group, since the order ofa is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. These special cases wereknown long before the general theorem was proved.The theorem also shows that any group of prime order is cyclic and simple. This in turn can be used to proveWilson's theorem, that if p is prime then p is a factor of (p-1)!+1.

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Lagrange's theorem (group theory) 156

Existence of subgroups of given orderLagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order ofsome subgroup. This does not hold in general: given a finite group G and a divisor d of |G|, there does notnecessarily exist a subgroup of G with order d. The smallest example is the alternating group G = A4 which has 12elements but no subgroup of order 6. A CLT group is a finite group with the property that for every divisor of theorder of the group, there is a subgroup of that order. It is known that a CLT group must be solvable and that everysupersolvable group is a CLT group: however there exists solvable groups which are not CLT and CLT groupswhich are not supersolvable.There are partial converses to Lagrange's theorem. For general groups, Cauchy's theorem guarantees the existence ofan element, and hence of a cyclic subgroup, of order any prime dividing the group order; Sylow's theorem extendsthis to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order. Forsolvable groups, Hall's theorems assert the existence of a subgroup of order equal to any unitary divisor of the grouporder (that is, a divisor coprime to its cofactor).

HistoryLagrange did not prove Lagrange's theorem in its general form. He stated, in his article Réflexions sur la résolutionalgébrique des équations,[1] that if a polynomial in n variables has its variables permuted in all n ! ways, the numberof different polynomials that are obtained is always a factor of n !. (For example if the variables x, y, and z arepermuted in all 6 possible ways in the polynomial x + y - z then we get a total of 3 different polynomials: x + y − z, x+ z - y, and y + z − x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetricgroup Sn of the subgroup H of permutations which preserve the polynomial. (For the example of x + y − z, thesubgroup H in S3 contains the identity and the transposition (xy).) So the size of H divides n !. With the laterdevelopment of abstract groups, this result of Lagrange on polynomials was recognized to extend to the generaltheorem about finite groups which now bears his name.Lagrange did not prove his theorem; all he did, essentially, was to discuss some special cases. The first completeproof of the theorem was provided by Abbati and published in 1803.[2]

Notes[1] Lagrange, J. L. (1771) "Réflexions sur la résolution algébrique des équations" [Reflections on the algebraic solution of equations] (part II),

Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin, pages 138-254; see especially pages 202-203. Availableon-line (in French, among Lagrange's collected works) at: http:/ / math-doc. ujf-grenoble. fr/ cgi-bin/ oeitem?id=OE_LAGRANGE__3_205_0[Click on "Section seconde. De la résolution des équations du quatrième degré 254-304"].

[2] P. Abbati (1803) "Lettera di Pietro Abbati Modenese al socio Paolo Ruffini da questo presentata il di 16. Décembre 1802" [Letter from PietroAbbati of Modena to the member Paolo Ruffini, who submitted it on the 16. December 1802], Memorie di Matematica e di Fisica dellaSocietà Italiana delle Scienze, vol. 10 (part 2), pages 385-409. See also: Richard L. Roth (April 2001) "A history of Lagrange's theorem ongroups," Mathematics Magazine, vol. 74, no. 2, pages 99-108.

References• Bray, Henry G. (1968), "A note on CLT groups", Pacific J. Math. 27 (2): 229–231• Gallian, Joseph (2006), Contemporary Abstract Algebra (6th ed.), Boston: Houghton Mifflin,

ISBN 978-0-618-51471-7• Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: John Wiley & Sons,

ISBN 978-0-471-43334-7, MR2286236• Roth, Richard R. (2001), "A History of Lagrange's Theorem on Groups", Mathematics Magazine 74 (2): 99–108,

doi:10.2307/2690624, JSTOR 2690624

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Lasker–Noether theoremIn mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means thatevery ideal can be written as an intersection of finitely many primary ideals (which are related to, but not quite thesame as, powers of prime ideals). The theorem was first proven by Emanuel Lasker (1905) for the special case ofpolynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether (1921).The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally thefundamental theorem of finitely generated abelian groups to all Noetherian rings.It has a straightforward extension to modules stating that every submodule of a finitely generated module over aNoetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case,considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primarydecomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and forthe special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finiteunion of (irreducible) varieties.The first algorithm for computing primary decompositions for polynomial rings was published by Noether's studentGrete Hermann (1926).

DefinitionsWrite R for a commutative ring, and M and N for modules over it.• A zero divisor of a module M is an element x of R such that xm = 0 for some non-zero m in M.• An element x of R is called nilpotent in M if xnM = 0 for some positive integer n.• A module is called coprimary if every zero divisor of M is nilpotent in M. For example, groups of prime power

order and free abelian groups are coprimary modules over the ring of integers.• A submodule M of a module N is called a primary submodule if N/M is coprimary.• An ideal I is called primary if it is a primary submodule of R. This is equivalent to saying that if ab is in I then

either a is in I or bn is in I for some n, and to the condition that every zero-divisor of the ring R/I is nilpotent.• A submodule M of a module N is called irreducible if it is not an intersection of two strictly larger submodules.• An associated prime of a module M is a prime ideal that is the annihilator of some element of M.

StatementThe Lasker–Noether theorem for modules states every submodule of a finitely generated module over a Noetherianring is a finite intersection of primary submodules. For the special case of ideals it states that every ideal of aNoetherian ring is a finite intersection of primary ideals.An equivalent statement is: every finitely generated module over a Noetherian ring is contained in a finite product ofcoprimary modules.The Lasker–Noether theorem follows immediately from the following three facts:• Any submodule of a finitely generated module over a Noetherian ring is an intersection of a finite number of

irreducible submodules.• If M is an irreducible submodule of a finitely generated module N over a Noetherian ring then N/M has only one

associated prime ideal.• A finitely generated module over a Noetherian ring is coprimary if and only if it has at most one associated prime.

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Irreducible decomposition in ringsThe study of the decomposition of ideals in rings began as a remedy for the lack of unique factorization in numberfields like

,in which

.If a number does not factor uniquely into primes, then the ideal generated by the number may still factor into theintersection of powers of prime ideals. Failing that, an ideal may at least factor into the intersection of primaryideals.Let R be a Noetherian ring, and I an ideal in R. Then I has an irredundant primary decomposition into primary ideals.

Irredundancy means:

• Removing any of the changes the intersection, i.e.,

for all i, where the hat denotes omission.

• The associated prime ideals are distinct.

More over, this decomposition is unique in the following sense: the set of associated prime ideals is unique, and theprimary ideal above every minimal prime in this set is also unique. However, primary ideals which are associatedwith non-minimal prime ideals are in general not unique.In the case of the ring of integers , the Lasker–Noether theorem is equivalent to the fundamental theorem ofarithmetic. If an integer n has prime factorization , then the primary decomposition of the idealgenerated by , is

Minimal decompositions and uniquenessIn this section, all modules will be finitely generated over a Noetherian ring R.A primary decomposition of a submodule M of a module N is called minimal if it has the smallest possible numberof primary modules. For minimal decompositions, the primes of the primary modules are uniquely determined: theyare the associated primes of N/M. Moreover the primary submodules associated to the minimal or isolatedassociated primes (those not containing any other associated primes) are also unique. However the primarysubmodules associated to the non-minimal associated primes (called embedded primes for geometric reasons) neednot be unique.Example: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, y2). Then M has two different minimalprimary decompositions M = (y) ∩ (x, y2) = (y) ∩ (x + y, y2). The minimal prime is (y) and the embedded prime is(x, y).

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When the conclusion does not holdThe decomposition does not hold in general for non-commutative Noetherian rings. Noether gave an example of anon-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.

Additive theory of idealsThis result is the first in an area now known as the additive theory of ideals, which studies the ways of representingan ideal as the intersection of a special class of ideals. The decision on the "special class", e.g., primary ideals, is aproblem in itself. In the case of non-commutative rings, the class of tertiary ideals is a useful substitute for the classof primary ideals.

References• Danilov, V.I. (2001), "Lasker ring" [1], in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer,

ISBN 978-1556080104• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York:

Springer-Verlag, ISBN 978-0-387-94268-1; 978-0-387-94269-8, MR1322960, esp. section 3.3.• Hermann, Grete (1926), "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale", Mathematische

Annalen 95: 736–788, doi:10.1007/BF01206635. English translation in Communications in Computer Algebra32/3 (1998): 8–30.

• Lasker, E. (1905), "Zur Theorie der Moduln und Ideale", Math. Ann. 60: 19–116, doi:10.1007/BF01447495• Markov, V.T. (2001), "Primary decomposition" [2], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,

Springer, ISBN 978-1556080104• Noether, Emmy (1921), "Idealtheorie in Ringbereichen" [3], Mathematische Annalen 83 (1): 24,

doi:10.1007/BF01464225• Curtis, Charles (1952), "On Additive Ideal Theory in General Rings", American Journal of Mathematics (The

Johns Hopkins University Press) 74 (3): 687–700, doi:10.2307/2372273, JSTOR 2372273• Krull, Wolfgang (1928), [year=1928 "Zur Theorie der zweiseitigen Ideale in nichtkommutativen Bereichen"],

Mathematische Zeitschrift 28 (1): 481–503, doi:10.1007/BF01181179, year=1928

References[1] http:/ / eom. springer. de/ L/ l057600. htm[2] http:/ / eom. springer. de/ P/ p074450. htm[3] http:/ / www. springerlink. com/ content/ m3457w8h62475473/ fulltext. pdf

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Latimer-MacDuffee theoremThe Latimer-MacDuffee theorem is a theorem in abstract algebra, a branch of mathematics.

Let be a monic, irreducible polynomial of degree . The Latimer-MacDuffee theorem gives a one-to-onecorrespondence between -similarity classes of matrices with characteristic polynomial and the idealclasses in the order

where ideals are considered equivalent if they are equal up to an overall (nonzero) rational scalar multiple. (Note thatthis order need not be the full ring of integers, so nonzero ideals need not be invertible.) Since an order in a numberfield has only finitely many ideal classes (even if it is not the maximal order, and we mean here ideals classes for allnonzero ideals, not just the invertible ones), it follows that there are only finitely many conjugacy classes of matricesover the integers with characteristic polynomial .

References• A Correspondence Between Classes of Ideals and Classes of Matrices, by Claiborne G. Latimer; C. C. MacDuffee

The Annals of Mathematics. 1933.

Lattice theoremIn mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or thecorrespondence theorem, states that if is a normal subgroup of a group , then there exists a bijection fromthe set of all subgroups of such that contains , onto the set of all subgroups of the quotient group

. The structure of the subgroups of is exactly the same as the structure of the subgroups of containing with collapsed to the identity element.This establishes a monotone Galois connection between the lattice of subgroups of and the lattice of subgroupsof , where the associated closure operator on subgroups of is Specifically, If

G is a group,N is a normal subgroup of G,

is the set of all subgroups A of G such that , andis the set of all subgroups of G/N,

then there is a bijective map such that

for all One further has that if A and B are in , and A' = A/N and B' = B/N, then

• if and only if ;• if then , where B:A is the index of A in B (the number of cosets bA of A in B);• where is the subgroup of generated by • and• is a normal subgroup of if and only if is a normal subgroup of This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under thebijection onto subgroups of a quotient group.

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References• W.R. Scott: Group Theory, Prentice Hall, 1964.

Levitzky's theoremIn mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after JacobLevitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent.[1] [2] Levitzky'stheorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution toone of Köthe's questions as described in (Levitzki 1945). The result was originally submitted in 1939 as (Levitzki1950), and a particularly simple proof was given in (Utumi 1963).

Notes[1] Herstein, Theorem 1.4.5, p. 37[2] Isaacs, Theorem 14.38, p. 210

References• I. Martin Isaacs (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company,

ISBN 0-534-19002-2• I.N. Herstein (1968), Noncommutative rings (1st ed.), The Mathematical Association of America,

ISBN 0-88385-015-X• J. Levitzki (1950). "On multiplicative systems" (http:/ / www. numdam. org/ item?id=CM_1951__8__76_0).

Compositio Math. 8: 76–80. MR0033799.• Levitzki, Jakob (1945), "Solution of a problem of G. Koethe", American Journal of Mathematics (The Johns

Hopkins University Press) 67 (3): 437–442, doi:10.2307/2371958, ISSN 0002-9327, JSTOR 2371958,MR0012269

• Utumi, Yuzo (1963), "Mathematical Notes: A Theorem of Levitzki", The American Mathematical Monthly(Mathematical Association of America) 70 (3): 286, doi:10.2307/2313127, ISSN 0002-9890, JSTOR 2313127,MR1532056

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Lie's third theoremIn mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g,over the real numbers, is the Lie algebra associated to some Lie group G. The relationship to the history has thoughbecome confused.There were (naturally) two other preceding theorems, of Sophus Lie. Those relate to the infinitesimaltransformations of a transformation group acting on a smooth manifold. But, in fact, that language is anachronistic.The manifold concept was not clearly defined at the time, the end of the nineteenth century, when Lie was foundingthe theory. The conventional third theorem on the list was a result stating the Jacobi identity for the infinitesimaltransformations, of a local Lie group. This result has a converse, stating that in the presence of a Lie algebra ofvector fields, integration gives a local Lie group action. The result initially stated is an intrinsic and global converseto the original theorem, therefore.

External links• Encyclopaedia of Mathematics (EoM) article at Springer.de [1]

References[1] http:/ / eom. springer. de/ l/ l058760. htm

Lie–Kolchin theoremIn mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie'stheorem is the analog for linear Lie algebras.It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and

a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L ofV such that

That is, ρ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation. This isequivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all

.Because every (nonzero finite-dimensional) representation of G has a one-dimensional invariant subspace accordingto the Lie–Kolchin theorem, every irreducible finite-dimensional representation of a connected and solvable linearalgebraic group G has dimension one, which is another way to state the Lie–Kolchin theorem.Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector spaceover an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace.The result for Lie algebras was proved by Sophus Lie (1876) and for algebraic groups was proved by EllisKolchin (1948, p.19).The Borel fixed point theorem generalizes the Lie–Kolchin theorem.

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TriangularizationSometimes the theorem is also referred to as the Lie–Kolchin triangularization theorem because by induction itimplies that with respect to a suitable basis of V the image has a triangular shape; in other words, the imagegroup is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices,the standard Borel subgroup of GL(n,K): the image is simultaneously triangularizable.The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.

Lie's theoremLie's theorem states that if V is a finite dimensional vector space over an algebraically closed field of characteristic 0,then for any solvable Lie algebra of endomorphisms of V there is a vector that is an eigenvector for every element ofthe Lie algebra.Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra arerepresented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matricesare simultaneously upper triangularizable, as commuting matrices form an abelian Lie algebra, which is a fortiorisolvable.A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 hasa nilpotent derived algebra.

Counter-examplesIf the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complexnumbers of absolute value one is a one-dimensional commutative (and thereforesolvable) linear algebraic group over the real numbers which has a two-dimensional representation into the specialorthogonal group SO(2) without an invariant (real) line. Here the image of is the orthogonalmatrix

For algebraically closed fields of characteristic p>0 Lie's theorem holds provided the dimension of the representationis less than p, but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Liealgebra spanned by 1, x, and d/dx acting on the p-dimensional vector space k[x]/(xp), which has no eigenvectors.Taking the semidirect product of this 3-dimensional Lie algebra by the p-dimensional representation (considered asan abelian Lie algebra) gives a solvable Lie algebra whose derived algebra is not nilpotent.

References• Gorbatsevich, V.V. (2001), "Lie–Kolchin theorem" [1], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,

Springer, ISBN 978-1556080104• Kolchin, E. R. (1948), "Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary

differential equations", Annals of Mathematics. Second Series 49: 1–42, ISSN 0003-486X, JSTOR 1969111,MR0024884

• Lie, Sophus (1876), "Theorie der Transformationsgruppen. Abhandlung II" [2], Archiv for Mathematik OgNaturvidenskab 1: 152–193

• William C. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics vol. 66, SpringerVerlag New York, 1979 (chapter 10, in particular section 10.2).

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References[1] http:/ / eom. springer. de/ l/ l058710. htm[2] http:/ / www. archive. org/ details/ archivformathem02sarsgoog

Maschke's theoremIn mathematics, Maschke's theorem,[1] [2] named after Heinrich Maschke,[3] is a theorem in group representationtheory that concerns the decomposition of representations of a finite group into irreducible pieces. If (V, ρ) is afinite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariantsubspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, therepresentation (V, ρ) is completely reducible. More generally, the theorem holds for fields of positive characteristicp, such as the finite fields, if the prime p doesn't divide the order of G.

Reformulation and the meaningOne of the approaches to representations of finite groups is through module theory. Representations of a group G arereplaced by modules over its group algebra KG. Irreducible representations correspond to simple modules.Maschke's theorem addresses the question: is a general (finite-dimensional) representation built from irreduciblesubrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary modulesemisimple? In this context, the theorem can be reformulated as follows:

Let G be a finite group and K a field whose characteristic does not divide the order of G. Then KG, the groupalgebra of G, is a semisimple algebra.[4] [5]

The importance of this result stems from the well developed theory of semisimple rings, in particular, theArtin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When K is the field ofcomplex numbers, this shows that the algebra KG is a product of several copies of complex matrix algebras, one foreach irreducible representation.[6] If the field K has characteristic zero, but is not algebraically closed, for example, Kis a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra KG is aproduct of matrix algebras over division rings over K. The summands correspond to irreducible representations of Gover K.[7]

Returning to representation theory, Maschke's theorem and its module-theoretic version allow one to make generalconclusions about representations of a finite group G without actually computing them. They reduce the task ofclassifying all representations to a more manageable task of classifying irreducible representations, since when thetheorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from theJordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not beunique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over afield of characteristic zero is determined up to isomorphism by its character.

Notes[1] Maschke, Heinrich (1898-07-22). "Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer

Substitutionsgruppen [On the arithmetical character of the coefficients of the substitutions of finite linear substitution groups]" (http:/ /resolver. sub. uni-goettingen. de/ purl?GDZPPN002256975) (in German). Math. Ann. 50 (4): 492–498. JFM 29.0114.03. MR1511011. .

[2] Maschke, Heinrich (1899-07-27). "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einigedurchgehends verschwindende Coefficienten auftreten, intransitiv sind [Proof of the theorem that those finite linear substitution groups, inwhich some everywhere vanishing coefficients appear, are intransitive]" (http:/ / resolver. sub. uni-goettingen. de/ purl?GDZPPN002257599)(in German). Math. Ann. 52 (2–3): 363–368. JFM 30.0131.01. MR1511061. .

[3] O'Connor, John J.; Robertson, Edmund F., "Heinrich Maschke" (http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Maschke. html),MacTutor History of Mathematics archive, University of St Andrews, .

[4] It follows that every module over KG is a semisimple module.

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[5] The converse statement also holds: if the characteristic of the field divides the order of the group (the modular case), then the group algebra isnot semisimple.

[6] The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.[7] One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real

numbers but not over the complex numbers.

References• Lang, Serge (2002-01-08). Algebra. Graduate Texts in Mathematics, 211 (Revised 3rd ed.). New York:

Springer-Verlag. ISBN 978-0-387-95385-4. MR1878556. Zbl 0984.00001.• Serre, Jean-Pierre (1977-09-01). Linear Representations of Finite Groups. Graduate Texts in Mathematics, 42.

New York–Heidelberg: Springer-Verlag. ISBN 978-0-387-90190-9. MR0450380. Zbl 0355.20006.

Milnor conjectureIn mathematics, the Milnor conjecture was a proposal by John Milnor (1970) of a description of the MilnorK-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (or equivalentlyétale) cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky (1996, 2003a, 2003b).

Statement of the theoremLet F be a field of characteristic different from 2. Then there is an isomorphism

for all n ≥ 0.

About the proofThe proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Andrei Suslin,Fabien Morel, Eric Friedlander, and others, including the newly-minted theory of motivic cohomology (a kind ofsubstitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra.

GeneralizationsThe analogue of this result for primes other than 2 was known as the Bloch–Kato conjecture. Work of Voevodsky,Markus Rost, and Charles Weibel yielded a complete proof of this conjecture in 2009; the result is now called thenorm residue isomorphism theorem.

References• Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology [1], Clay

Mathematics Monographs, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1;978-0-8218-3847-1, MR2242284

• Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9: 318–344,doi:10.1007/BF01425486, ISSN 0020-9910, MR0260844

• Voevodsky, V. (1996), The Milnor Conjecture [2], Preprint• Voevodsky, Vladimir (2003a), "Reduced power operations in motivic cohomology" [3], Institut des Hautes Études

Scientifiques. Publications Mathématiques (98): 1–57, doi:10.1007/s10240-003-0009-z, ISSN 0073-8301,MR2031198

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• Voevodsky, Vladimir (2003b), "Motivic cohomology with Z/2-coefficients" [4], Institut des Hautes ÉtudesScientifiques. Publications Mathématiques (98): 59–104, doi:10.1007/s10240-003-0010-6, ISSN 0073-8301,MR2031199

References[1] http:/ / math. rutgers. edu/ ~weibel/ motiviclectures. html[2] http:/ / www. math. uiuc. edu/ K-theory/ 0170[3] http:/ / www. numdam. org/ item?id=PMIHES_2003__98__1_0[4] http:/ / www. numdam. org/ item?id=PMIHES_2003__98__59_0

Mordell–Weil theoremIn mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the groupA(K) of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group. The case with Aan elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posedby Poincaré around 1908; it was proved by Louis Mordell in 1922.The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as theseventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded inestablishing the finiteness of the quotient group E(Q)/2E(Q) which forms a major step in the proof. Certainly thefiniteness of this group is a necessary condition for E(Q) to be finitely-generated; and it shows that the rank is finite.This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point on E.Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curvesover arbitrary number fields in his doctoral dissertation published in 1928. More abstract methods were required, tocarry out a proof with the same basic structure. The second half of the proof needs some type of height function, interms of which to bound the 'size' of points of A(K). Some measure of the co-ordinates will do; heights arelogarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set ofhomogeneous coordinates. For an abelian variety, there is no a priori preferred representation, though, as aprojective variety.Both halves of the proof have been improved significantly, by subsequent technical advances: in Galois cohomologyas applied to descent, and in the study of the best height functions (which are quadratic forms). The theorem leftunanswered a number of questions:• Calculation of the rank (still a demanding computational problem, and not always effective, as far as it is

currently known).• Meaning of the rank: see Birch and Swinnerton-Dyer conjecture.• For a curve C in its Jacobian variety as A, can the intersection of C with A(K) be infinite? (Not unless C = A,

according to Mordell's conjecture, proved by Faltings.)• In the same context, can C contain infinitely many torsion points of A? (No, according to the Manin-Mumford

conjecture proved by Raynaud, other than in the elliptic curve case.)

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References• A. Weil, L'arithmétique sur les courbes algébriques, Acta Math 52, (1929) p. 281-315, reprinted in vol 1 of his

collected papers ISBN 0387903305• L.J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc Cam.

Phil. Soc. 21, (1922) p. 179.• J. H. Silverman, The arithmetic of elliptic curves, ISBN 0387962034 second edition

Multinomial theoremIn mathematics, the multinomial theorem says how to write a power of a sum in terms of powers of the terms inthat sum. It is the generalization of the binomial theorem to polynomials.

TheoremFor any positive integer m and any nonnegative integer n, the multinomial formula tells us how a polynomialexpands when raised to an arbitrary power:

The sum is taken over all sequences of nonnegative integer indices k1 through km such the sum of all ki is n. That is,for each term in the expansion, the exponents must add up to n. Also, as with the binomial theorem, quantities of theform x0 that appear are taken to equal 1 (even when x equals zero). Alternatively, this can be written concisely usingmultiindices as

where α = (α1,α2,…,αm) and xα = x1α1x2

α2⋯xmαm.

Number of multinomial coefficientsThe number of terms in multinomial sum, #n,m, is equal to the number of monomials of degree n on the variablesx1, …, xm:

The count can be easily performed using the Stars and bars (combinatorics) method.

Multinomial coefficientsThe numbers

(which can also be written as:)

are the multinomial coefficients. Just like "n choose k" are the coefficients when you raise a binomial to the nth

power (e.g. the coefficients are 1,3,3,1 for (a + b)3, where n = 3), the multinomial coefficients appear when oneraises a multinomial to the nth power (e.g. (a + b + c)3)

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Sum of all multinomial coefficients

The substitution of xi = 1 for all i into:

gives immediately that

Central multinomial coefficients

All of the multinomial coefficients for which the following holds true:

are central multinomial coefficients: the greatest ones and all of equal size.A special case for m = 2 is central binomial coefficient.

Example multinomial coefficients

We could have calculated each coefficient by first expanding (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac, thenself-multiplying it again to get (a + b + c)3 (and then if we were raising it to higher powers, we'd multiply it by itselfeven some more). However this process is slow, and can be avoided by using the multinomial theorem. Themultinomial theorem "solves" this process by giving us the closed form for any coefficient we might want. It ispossible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. Forexample:

has the coefficient

has the coefficient .

We could have also had a 'd' variable, or even more variables—hence the multinomial theorem.The binomial theorem and binomial coefficients are special cases, for m = 2, of the multinomial theorem andmultinomial coefficients, respectively.

Interpretations

Ways to put objects into boxesThe multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing ndistinct objects into m distinct bins, with k1 objects in the first bin, k2 objects in the second bin, and so on.[1]

Number of ways to select according to a distributionIn statistical mechanics and combinatorics if one has a number distribution of labels then the multinomialcoefficients naturally arise from the binomial coefficients. Given a number distribution {ni} on a set of N total items,ni represents the number of items to be given the label i. (In statistical mechanics i is the label of the energy state.)The number of arrangements is found by

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• Choosing n1 of the total N to be labeled 1. This can be done ways.

• From the remaining N − n1 items choose n2 to label 2. This can be done ways.

• From the remaining N − n1 − n2 items choose n3 to label 3. Again, this can be done ways.

Multiplying the number of choices at each step results in:

Upon cancellation, we arrive at the formula given in the introduction.

Number of unique permutations of wordsIn addition, the multinomial coefficient is also the number of distinct ways to permute a multiset of n elements, andki are the multiplicities of each of the distinct elements. For example, the number of distinct permutations of theletters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps is

(This is just like saying that there are 11! ways to permute the letters—the common interpretation of factorial as thenumber of unique permutations. However, we created duplicate permutations, due to the fact that some letters are thesame, and must divide to correct our answer.)

Generalized Pascal's triangleOne can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex. Thisprovides a quick way to generate a lookup table for multinomial coefficients.The case of n = 3 can be easily drawn by hand. The case of n = 4 can be drawn with effort as a series of growingpyramids.

ProofThis proof of the multinomial theorem uses the binomial theorem and induction on m.First, for m = 1, both sides equal x1

n since there is only one term k1 = n in the sum. For the induction step, supposethe multinomial theorem holds for m. Then

by the induction hypothesis. Applying the binomial theorem to the last factor,

which completes the induction. The last step follows because

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as can easily be seen by writing the three coefficients using factorials as follows:

References[1] National Institute of Standards and Technology (May, 11, 2010). "NIST Digital Library of Mathematical Functions" (http:/ / dlmf. nist. gov/

). Section 26.4 (http:/ / dlmf. nist. gov/ 26. 4). . Retrieved August 30, 2010.

Nielsen–Schreier theoremIn group theory, a branch of mathematics, the Nielsen–Schreier theorem is the statement that every subgroup of afree group is itself free.[1] [2] [3] It is named after Jakob Nielsen and Otto Schreier.

Statement of the theoremA free group may be defined from a group presentation consisting of a set of generators and the empty set ofrelations (equations that the generators satisfy). That is, it is the unique group in which every element is a product ofsome sequence of generators and their inverses, and in which there are no equations between group elements that donot follow in a trivial way from the equations gg−1 describing the relation between a generator and its inverse. Theelements of a free group may be described as all of the possible reduced words formed by sequences of generatorsand their inverse that have no adjacent pair of a generator and the inverse of the same generator.The Nielsen–Schreier theorem states that if G is a subgroup of a free group, then G is itself isomorphic to a freegroup. That is, there exists a subset S of elements of G such that every element in S is a product of members of Sand their inverses, and such that S satisfies no nontrivial relations.

ExampleLet G be the free group with two generators, a and b, and let E be the subgroup consisting of all reduced wordsthat are products of evenly many generators or their inverses. Then E is itself generated by the six elements p = aa,q = ab, r = ab−1, s = ba, t = ba−1, and u = bb. A factorization of any reduced word in E into these generators andtheir inverses may be constructed simply by taking consecutive pairs of symbols in the reduced word. However, thisis not a free presentation of E because it satisfies the relations p = qr−1 = rq−1 and s = tu−1 = ut−1. Instead, E isgenerated as a free group by the three elements p = aa, q = ab, and s = ba. Any factorization of a word into a productof generators from the six-element generating set {p, q, r, s, t, u} can be transformed into a product of generatorsfrom this smaller set by replacing r with ps−1, replacing t with sp−1, and replacing u with sp−1q. There are noadditional relations satisfied by these three generators, so E is the free group generated by p, q, and s.[4] TheNielsen–Scheier theorem states that this example is not a coincidence: like E, every subgroup of a free group can begenerated as a free group, possibly with a larger set of generators.

ProofIt is possible to prove the Nielsen–Scheier theorem using topology.[1] A free group G on a set of generators is the fundamental group of a bouquet of circles, a topological graph with a single vertex and with an edge for each generator.[5] Any subgroup H of the fundamental group is itself a fundamental group of a covering space of the bouquet, a (possibly infinite) topological Schreier coset graph that has one vertex for each coset of the subgroup.[6]

And in any topological graph, it is possible to shrink the edges of a spanning tree of the graph, producing a bouquet of circles that has the same fundamental group H. Since H is the fundamental group of a bouquet of circles, it is

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itself free.[5]

According to Schreier's subgroup lemma, a set of generators for a free presentation of H may be constructed fromcycles in the covering graph formed by concatenating a spanning tree path from a base point (the coset of theidentity) to one of the cosets, a single non-tree edge, and an inverse spanning tree path from the other endpoint of theedge back to the base point.[7]

Axiomatic foundationsAlthough several different proofs of the Nielsen–Schreier theorem are known, they all depend on the axiom ofchoice. In the proof based on fundamental groups of bouquets, for instance, the axiom of choice appears in the guiseof the statement that every connected graph has a spanning tree. The use of this axiom is necessary, as there existmodels of Zermelo–Fraenkel set theory in which the axiom of choice and the Nielsen–Schreier theorem are bothfalse. The Nielsen–Schreier theorem in turn implies a weaker version of the axiom of choice, for finite sets.[8]

HistoryThe Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroupof a free abelian group is free abelian.[3]

Jakob Nielsen (1921) originally proved a restricted form of the theorem, stating that any finitely-generated subgroupof a free group is free. His proof involves performing a sequence of Nielsen transformations on the subgroup'sgenerating set that reduce their length (as reduced words in the free group from which they are drawn).[1] [9] OttoSchreier proved the Nielsen–Schreier theorem in its full generality in his 1926 habilitation thesis, Die Untergruppender freien Gruppe, also published in 1927 in Abh. math. Sem. Hamburg. Univ.[10] [11]

The topological proof based on fundamental groups of bouquets of circles is due to Reinhold Baer and FriedrichLevi (1936). Another topological proof, based on the Bass–Serre theory of group actions on trees, was published byJean-Pierre Serre (1970).[12]

Notes[1] Stillwell (1993), Section 2.2.4, The Nielsen–Schreier Theorem, pp. 103–104.[2] Magnus Solitar Karass (http:/ / v3. espacenet. com/ textdoc?DB=EPODOC& IDX=MagnusKarass), Corollary 2.9, p. 95.[3] Johnson (1980), Section 2, The Nielsen–Schreier Theorem, pp. 9–23.[4] Johnson (1997), ex. 15, p. 12.[5] Stillwell (1993), Section 2.1.8, Freeness of the Generators, p. 97.[6] Stillwell (1993), Section 2.2.2, The Subgroup Property, pp. 100–101.[7] Stillwell (1993), Section 2.2.6, Schreier Transversals, pp. 105–106.[8] Läuchli (1962); Howard (1985).[9] Magnus Solitar Karass (http:/ / v3. espacenet. com/ textdoc?DB=EPODOC& IDX=MagnusKarass), Section 3.2, A Reduction Process, pp.

121–140.[10] O'Connor, John J.; Robertson, Edmund F., "Nielsen–Schreier theorem" (http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/

Schreier. html), MacTutor History of Mathematics archive, University of St Andrews, .[11] Hansen, Vagn Lundsgaard (1986), Jakob Nielsen, Collected Mathematical Papers: 1913-1932, Birkhäuser, p. 117, ISBN 9780817631406.[12] Rotman (1995), The Nielsen–Schreier Theorem, pp. 383–387.

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References• Baer, Reinhold; Levi, Friedrich (1936), "Freie Produkte und ihre Untergruppen", Compositio Math. 3: 391–398.• Howard, Paul E. (1985), "Subgroups of a free group and the axiom of choice", The Journal of Symbolic Logic 50

(2): 458–467, doi:10.2307/2274234, MR793126.• Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note

series, 42, Cambridge University Press, ISBN 9780521231084.• Johnson, D. L. (1997), Presentations of Groups, London Mathematical Society student texts, 15 (2nd ed.),

Cambridge University Press, ISBN 9780521585422.• Läuchli, Hans (1962), "Auswahlaxiom in der Algebra", Commentarii Mathematici Helvetici 37: 1–18,

MR0143705.• Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (1976), Combinatorial Group Theory (2nd revised ed.),

Dover Publications.• Nielsen, Jakob (1921), "Om regning med ikke-kommutative faktorer og dens anvendelse i gruppeteorien" (in

Danish), Math. Tidsskrift B, 1921: 78–94, JFM 48.0123.03.• Rotman, Joseph J. (1995), An Introduction to the Theory of Groups, Graduate Texts in Mathematics, 148 (4th

ed.), Springer-Verlag, ISBN 9780387942858.• Serre, J.-P. (1970), Groupes Discretes, Extrait de I'Annuaire du College de France, Paris.• Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, 72

(2nd ed.), Springer-Verlag.

Perron–Frobenius theoremIn linear algebra, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912),asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the correspondingeigenvector has strictly positive components, and also asserts a similar statement for certain classes of nonnegativematrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theoryof dynamical systems (subshifts of finite type); to economics (Leontief's input-output model[1] 8.3.6 p. 681 [2]); todemography (Leslie population age distribution model[1] 8.3.7 p. 683 [2]); to mathematical background of the internetsearch engines[3] 15.2 p. 167 [4] and even to ranking of football teams[5] p. 80 [6]."In addition to saying something useful, the Perron–Frobenius theory is elegant. It is a testament to the fact thatbeautiful mathematics eventually tends to be useful, and useful mathematics eventually tends to be beautiful.[1] "

Statement of the Perron–Frobenius theoremA matrix in which all entries are positive real numbers is here called positive and a matrix whose entries arenon-negative real numbers is here called non-negative. The eigenvalues of a real square matrix A are complexnumbers and collectively they make up the spectrum of the matrix. The exponential growth rate of the matrix powersAk as k → ∞ is controlled by the eigenvalue of A with the largest absolute value. The Perron–Frobenius theoremdescribes the properties of the leading eigenvalue and of the corresponding eigenvectors when A is a non-negativereal square matrix. Early results were due to Oskar Perron (1907) and concerned positive matrices. Later, GeorgFrobenius (1912) found their extension to certain classes of non-negative matrices.

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Positive matricesLet A = (aij) be an n × n positive matrix: aij > 0 for 1 ≤ i, j ≤ n. Then the following statements hold.1. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue, such that r is an

eigenvalue of A and any other eigenvalue λ (possibly, complex) is strictly smaller than r in absolute value, |λ| < r.Thus, the spectral radius ρ(A) is equal to r.

2. The Perron–Frobenius eigenvalue is simple: r is a simple root of the characteristic polynomial of A.Consequently, the eigenspace associated to r is one-dimensional. (The same is true for the left eigenspace, i.e., theeigenspace for AT.)

3. There exists an eigenvector v = (v1,…,vn) of A with eigenvalue r such that all components of v are positive: A v =r v, vi > 0 for 1 ≤ i ≤ n. (Respectively, there exists a positive left eigenvector w : wT A = r wT, wi > 0.)

4. There are no other positive (moreover non-negative) eigenvectors except positive multiples of v (respectively, lefteigenvectors except w), i.e. all other eigenvectors must have at least one negative or non-real component.

5. , where the left and right eigenvectors for A are normalized so that wTv = 1. Moreover, the

matrix v wT is the projection onto the eigenspace corresponding to r. This projection is called the Perronprojection.

6. Collatz–Wielandt formula: for all non-negative non-zero vectors x, let f(x) be the minimum value of [Ax]i / xitaken over all those i such that xi ≠ 0. Then f is a real valued function whose maximum is the Perron–Frobeniuseigenvalue.

7. A "Min-max" Collatz–Wielandt formula takes a form similar to the one above: for all strictly positive vectors x,let g(x) be the maximum value of [Ax]i / xi taken over i. Then g is a real valued function whose minimum is thePerron–Frobenius eigenvalue.

8. The Perron–Frobenius eigenvalue satisfies the inequalities

These claims can be found in Meyer[1] chapter 8 [2] claims 8.2.11-15 page 667 and exercises 8.2.5,7,9 pages668-669.The left and right eigenvectors v and w are usually normalized so that the sum of their components is equal to 1; inthis case, they are sometimes called stochastic eigenvectors.

Non-negative matricesAn extension of the theorem to matrices with non-negative entries is also available. In order to highlight thesimilarities and differences between the two cases the following points are to be noted: every non-negative matrixcan be obviously obtained as a limit of positive matrices, thus one obtains the existence of an eigenvector withnon-negative components; obviously the corresponding eigenvalue will be non-negative and greater or equal inabsolute value than all other eigenvalues (see Meyer[1] chapter 8.3 [2] page 670 or Gantmacher[7] chapter XIII.3theorem 3 page 66 [8] for details). However, the simple examples

show that for non-negative matrices there may exist eigenvalues of the same absolute value as the maximal one ((1)and (−1) – eigenvalues of the first matrix); moreover the maximal eigenvalue may not be a simple root of thecharacteristic polynomial, can be zero and the corresponding eigenvector (1,0) is not strictly positive (secondexample). So it may seem that most properties are broken for non-negative matrices, however Frobenius found theright way.The key feature of theory in the non-negative case is to find some special subclass of non-negative matrices – irreducible matrices – for which a non-trivial generalization is possible. Namely, although eigenvalues attaining

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maximal absolute value may not be unique, the structure of maximal eigenvalues is under control: they have theform ei2πl/hr, where h is some integer number – period of matrix, r is a real strictly positive eigenvalue,l = 0, 1, ..., h − 1. The eigenvector corresponding to r has strictly positive components (in contrast with the generalcase of non-negative matrices, where components are only non-negative). Also all such eigenvalues are simple rootsof the characteristic polynomial. Further properties are described below.

Classification of matrices

Let A be a square matrix (not necessarily positive or even real). The matrix A is irreducible if any of the followingequivalent properties holds.Definition 1 : A does not have non-trivial invariant coordinate subspaces. Here a non-trivial coordinate subspacemeans a linear subspace spanned by any proper subset of basis vectors. More explicitly, for any linear subspacespanned by basis vectors ei1 , ..., eik, n > k > 0 its image under the action of A is not contained in the same subspace.Definition 2: A cannot be conjugated into block upper triangular form by a permutation matrix P:

where E and G are non-trivial (i.e. of size greater than zero) square matrices.If A is non-negative other definitions exist:Definition 3: For every pair of indices i and j, there exists a natural number m such that (Am)ij is not equal to 0.Definition 4: One can associate with a matrix A a certain directed graph GA. It has exactly n vertices, where n is sizeof A, and there is an edge from vertex i to vertex j precisely when Aij > 0. Then the matrix A is irreducible if and onlyif its associated graph GA is strongly connected.A matrix is reducible if it is not irreducible.Let A be non-negative. Fix an index i and define the period of index i to be the greatest common divisor of allnatural numbers m such that (Am)ii > 0. When A is irreducible, the period of every index is the same and is called theperiod of A. When A is also irreducible, the period can be defined as the greatest common divisor of the lengths ofthe closed directed paths in GA (see Kitchens[9] page 16). The period is also called the index of imprimitivity(Meyer[1] page 674) or the order of cyclicity.If the period is 1, A is aperiodic.A matrix A is primitive if it is non-negative and its mth power is positive for some natural number m (i.e. the same mworks for all pairs of indices). It can be proved that primitive matrices are the same as irreducible aperiodicnon-negative matrices.A positive square matrix is primitive and a primitive matrix is irreducible. All statements of the Perron–Frobeniustheorem for positive matrices remain true for primitive matrices. However, a general non-negative irreducible matrixA may possess several eigenvalues whose absolute value is equal to the spectral radius of A, so the statements needto be correspondingly modified. Actually the number of such eigenvalues is exactly equal to the period. Results fornon-negative matrices were first obtained by Frobenius in 1912.

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Perron–Frobenius theorem for irreducible matrices

Let A be an irreducible non-negative n × n matrix with period h and spectral radius ρ(A) = r. Then the followingstatements hold.1. The number r is a positive real number and it is an eigenvalue of the matrix A, called the Perron–Frobenius

eigenvalue.2. The Perron–Frobenius eigenvalue r is simple. Both right and left eigenspaces associated with r are

one-dimensional.3. A has a left eigenvector v with eigenvalue r whose components are all positive.4. Likewise, A has a right eigenvector w with eigenvalue r whose components are all positive.5. The only eigenvectors whose components are all positive are those associated with the eigenvalue r.6. Matrix A has exactly h (where h is the period) complex eigenvalues with absolute value r. Each of them is a

simple root of the characteristic polynomial and is the product of r with an hth root of unity.7. Let ω = 2π/h. Then the matrix A is similar to eiωA, consequently the spectrum of A is invariant under

multiplication by eiω (corresponding to the rotation of the complex plane by the angle ω).8. If h > 1 then there exists a permutation matrix P such that

where the blocks along the main diagonal are zero square matrices.9. Collatz–Wielandt formula: for all non-negative non-zero vectors x let f(x) be the minimum value of [Ax]i /xi taken over all those i such that xi ≠ 0. Then f is a real valued function whose maximum is thePerron–Frobenius eigenvalue.10. The Perron–Frobenius eigenvalue satisfies the inequalities

The matrix shows that the blocks on the diagonal may be of different sizes, the matrices Aj need not

be square, and h need not divide n.

Further propertiesLet A be an irreducible non-negative matrix, then:1. (1+A)n−1 is a positive matrix. (Meyer[1] claim 8.3.5 p. 672 [2]).2. Wielandt's theorem. If |B|<A, then ρ(B)≤ρ(A). If equality holds (i.e. if μ=ρ(A)eiφ is eigenvalue for B), then B = eiφ

D AD−1 for some diagonal unitary matrix D (i.e. diagonal elements of D equals to eiΘl, non-diagonal are zero).(Meyer[1] claim 8.3.11 p. 675 [2]).

3. If some power Aq is reducible, then it is completely reducible, i.e. for some permutation matrix P, it is true that:

, where Ai are irreducible matrices having the same maximal

eigenvalue. The number of these matrices d is the greatest common divisor of q and h, where h is period of A.(Gantmacher[7] section XIII.5 theorem 9).

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4. If c(x)=xn+ck1 xn-k1 +ck2 xn-k2 + ... + cks xn-ks is the characteristic polynomial of A in which the only nonzerocoefficients are listed, then the period of A equals to the greatest common divisor for k1, k2, ... , ks.(Meyer[1] page679 [2]).

5. Cesàro averages: where the left and right eigenvectors for A are normalized

so that wtv = 1. Moreover the matrix v wt is the spectral projection corresponding to r - Perron projection.(Meyer[1] example 8.3.2 p. 677 [2]).

6. Let r be the Perron-Frobenius eigenvalue, then the adjoint matrix for (r-A) is positive (Gantmacher[7] sectionXIII.2.2 page 62 [10]).

7. If A has at least one non-zero diagonal element, then A is primitive. (Meyer[1] example 8.3.3 p. 678 [2]).The following facts worth to be mentioned.• If 0 ≤ A < B, the rA ≤ rB, moreover, if A is irreducible, then the inequality is strict: rA < rB.One of the definitions of primitive matrix requires A to be non-negative and there exists m, such that Am is positive.One may one wonder how big m can be, depending on the size of A. The following answers this question.• Assume A is non-negative primitive matrix of size n, then An2-2n+2 is positive. Moreover there exists a matrix M

given below, such that Mk remains not positive (just non-negative) for all k< n2-2n+2, in particular(Mn2-2n+1)11=0.

(Meyer[1] chapter 8 [2] example 8.3.4 page 679 and exercise 8.3.9 p. 685).

ApplicationsNumerous books have been written on the subject of non-negative matrices, and Perron–Frobenius theory isinvariably a central feature. So there is a vast application area and the examples given below barely begin to scratchits surface.

Non-negative matricesThe Perron–Frobenius theorem does not apply directly to non-negative matrices. Nevertheless any reducible squarematrix A may be written in upper-triangular block form (known as the normal form of a reducible matrix)[11]

PAP−1 =

where P is a permutation matrix and each Bi is a square matrix that is either irreducible or zero. Now if A isnon-negative then so are all the Bi and the spectrum of A is just the union of their spectra. Therefore many of thespectral properties of A may be deduced by applying the theorem to the irreducible Bi.For example the Perron root is the maximum of the ρ(Bi). Whilst there will still be eigenvectors with non-negativecomponents it is quite possible that none of these will be positive.

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Stochastic matricesA row (column) stochastic matrix is a square matrix each of whose rows (columns) consists of non-negative realnumbers whose sum is unity. The theorem cannot be applied directly to such matrices because they need not beirreducible.If A is row-stochastic then the column vector with each entry 1 is an eigenvector corresponding to the eigenvalue 1,which is also ρ(A) by the remark above. It may not be the only eigenvalue on the unit circle: and the associatedeigenspace can be multi-dimensional. If A is row-stochastic and irreducible then the Perron projection is alsorow-stochastic and all its rows are equal.

Algebraic graph theoryThe theorem has particular use in algebraic graph theory. The "underlying graph" of a nonnegative n-square matrix isthe graph with vertices numbered 1, ..., n and arc ij if and only if Aij ≠ 0. If the underlying graph of such a matrix isstrongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the adjacency matrix ofa strongly connected graph is irreducible.[12] [13]

Finite Markov chainsThe theorem has a natural interpretation in the theory of finite Markov chains (where it is the matrix-theoreticequivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in termsof the transition matrix of the chain; see, for example, the article on the subshift of finite type).

Compact operatorsMore generally, it can be extended to the case of non-negative compact operators, which, in many ways, resemblefinite-dimensional matrices. These are commonly studied in physics, under the name of transfer operators, orsometimes Ruelle–Perron–Frobenius operators (after David Ruelle). In this case, the leading eigenvaluecorresponds to the thermodynamic equilibrium of a dynamical system, and the lesser eigenvalues to the decay modesof a system that is not in equilibrium. Thus, the theory offers a way of discovering the arrow of time in what wouldotherwise appear to be reversible, deterministic dynamical processes, when examined from the point of view ofpoint-set topology.[14]

Proof methodsA common thread in many proofs is the Brouwer fixed point theorem. Another popular method is that of Wielandt(1950). He used the Collatz–Wielandt formula described above to extend and clarify Frobenius's work (seeGantmacher[7] section XIII.2.2 page 54 [15] ). The proof based on the spectral theory can be found in[16] from whichpart of the arguments are borrowed.

Perron root is strictly maximal eigenvalue for positive (and primitive) matrices• If A is a positive (or more generally primitive) matrix, then there exists real positive eigenvalue r

(Perron-Frobenius eigenvalue or Perron root), which is strictly greater in absolute value than all othereigenvalues, hence r is the spectral radius of A.

Pay attention that claim is wrong for general non-negative irreducible matrices, which have h eigenvalues with thesame absolute eigenvalue as r, where h is the period of A.Proof. Consider first the case of positive matrices. Let A be a positive matrix, assume that its spectral radius ρ(A) = 1 (otherwise consider A/ρ(A)). Hence, as it is known, there exists an eigenvalue λ on the unit circle, and all the other eigenvalues are less or equal 1 in absolute value. Assume that that λ ≠ 1. Then there exists a positive integer m such that Am is a positive matrix and the real part of λm is negative. Let ε be half the smallest diagonal entry of Am and set

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T = Am − ε1 which is yet another positive matrix. Moreover if Ax = λx then Amx = λmx thus λm − ε is an eigenvalue ofT. Because of the choice of m this point lies outside the unit disk consequently ρ(T) > 1. On the other hand all theentries in T are positive and less than or equal to those in Am so by Gelfand's formula ρ(T) ≤ ρ(Am) ≤ ρ(A)m = 1. Thiscontradiction means that λ=1 and there can be no other eigenvalues on the unit circle. Proof for positive matrices isfinished .Absolutely the same arguments can be applied to the case of primitive matrices, after one just need to mention thefollowing simple lemma, which clarifies the properties of primitive matricesLemma: Consider a non-negative A. Assume there exists m, such that Am is positive, then Am+1, Am+2, Am+3,... areall positive.Indeed, Am+1= A Am, so it can have zero element only if some row of A is entirely zero, but in this case the same rowof Am will be zero. Lemma is proved.After lemma is established one applies the same arguments as above for primitive matrices to prove the main claim.

Power method and the positive eigenpair• for a positive (or more generally irreducible non-negative) matrix A the dominant eigenvector is real and

strictly positive (for non-negative A respectively non-negative)It can be argued by the power method, which states that for sufficiently generic (in the sense discussed below) matrixA the sequence of vectors bk+1=Abk / | Abk | converges to the eigenvector with the maximum eigenvalue. (Initialvector b0 can be chosen arbitrary except some measure zero set). Starting with a non-negative vector b0 one gets thesequence of non-negative vectors bk. Hence the limiting vector is also non-negative. By power method this limitingvector is the desired the dominant eigenvector for A, so assertion is proved. Corresponding eigenvalue is clearlynon-negative.To accomplish the proof two arguments should be added. First, the power method converges for matrices which doesnot have several eigenvalues of the same absolute value as the maximal one. The previous section argumentguarantees this.Second, is to ensure strict positivity of all of the components of the eigenvector for the case of irreducible matrices.This follows from the following simple fact, which is of independent interest:

Lemma: consider a positive (or more generally irreducible non-negative) matrix A. Assume v is anynon-negative eigenvector for A, then it is necessarily strictly positive and corresponding eigenvalue is alsostrictly positive.

Proof. Recall that one of the definitions of irreducibility for non-negative matrices is that for all indexes i,j thereexists m, such that (Am)ij is strictly positive. Consider a non-negative eigenvector v, assume that at least one of itscomponents say j-th is strictly positive. Then one can deduce that the corresponding eigenvalue is strictly positive,indeed, consider n such that (An)ii >0, hence: rnvi = Anvi >= (An)iivi >0. Hence r is strictly positive. In the samemanner one can deduce strict positivity of the eigenvector. To prove it, consider m, such that (Am)ij >0, hence: rmvj =(Amv)j >= (Am)ijvi >0, hence vj is strictly positive, i.e. eigenvector is strictly positive. Proof is finished.

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Multiplicity oneThe proof that the Perron-Frobenius eigenvalue is simple root of the characteristic polynomial is also elementary.The arguments here are close to that ones in Meyer[1] chapter 8 [2] page 665, where details can be found.

• Eigenspace associated to Perron-Frobenius eigenvalue r is one-dimensionalConsider strictly positive eigenvector v corresponding to r. Assume there exists another eigenvector w with the sameeigenvalue. (Vector w can be chosen to be real, because A and r are both real, so the null space of A-r has a basisconsisting of real vectors). Assume at least one of the components of w is positive (otherwise multiply w by -1).Consider maximal possible α such that u=v- α w is non-negative. Then clearly one of the components of u is zero,otherwise α is not maximum. Obviously vector u is an eigenvector. It is non-negative, hence by the lemma describedin the previous section non-negativity implies strict positivity for any eigenvector. On the other hand it was statedjust above that at least one component of u is zero. The contradiction implies, that w does not exist. The claim isproved.

• There is no Jordan cells corresponding to the Perron-Frobenius eigenvalue r and all other eigenvalues whichhas the same absolute value

The idea is the following: if there is a Jordan cell, then the infinity norm ||(A/r)k||∞ tends to infinity for k → ∞ , but itwill actually contradict the existence of the positive eigenvector.The details are the following. Assume r=1, otherwise consider A/r. Let v be Perron-Frobenius strictly positiveeigenvector, so Av=v, then:

So ||Ak||∞ is bounded for all k. Actually it gives another proof that there is no eigenvalues which has greater absolutevalue then Perron-Frobenius one. And it gives more. It contradicts the existence of the Jordan cell for any eigenvaluewhich has absolute value equal to 1 (in particular for the Perron-Frobenius one), because existence of the Jordan cellimplies that ||Ak||∞ is unbounded. For example for two by two matrix:

hence ||Jk||∞ = |k+λ| (for |λ|=1), so it tends to infinity when k does so. Since ||Jk|| = ||C-1 AkC ||, then || Ak || >= ||Jk||/ (||C−1|| || C ||), so it also tends to infinity. Obtained contradiction implies that there is no Jordan cells for thecorresponding eigenvalues. Proof is finished.Combining the two claims above together one gets:

• the Perron-Frobenius eigenvalue r is simple root of the characteristic polynomial.Actually in the case of non primitive matrices, there exists other eigenvalues which has the same absolute value as r.Same claim is true for them, but it requires more work.

No other non-negative eigenvectors• Consider positive (or more generally irreducible non-negative matrix) A. The Perron-Frobenius eigenvector is

the only (up to multiplication by constant) non-negative eigenvector for A.So other eigenvectors should contain negative, or complex components. The idea of proof is the following -eigenvectors for different eigenvalues should be orthogonal in some sense, however two positive eigenvectors cannotbe orthogonal, so they correspond the same eigenvalue, but eigenspace for the Perron-Frobenius is one dimensional.The formal proof goes as follows. Assume there exists an eigenpair (λ, y) for A, such that vector y is positive.Consider (r, x), where x - is the right Perron-Frobenius eigenvector for A (i.e. eigenvector for At). Then: r xt y = (xt A)y= xt (A y)=λ xt y, also xt y >0, so one has: r=λ. But eigenspace for the Perron-Frobenius eigenvalue r is onedimensional, as it was established before. Hence non-negative eigenvector y is multiple of the Perron-Frobenius one.Assertion is proved.

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One can consult Meyer[1] chapter 8 [2] claim 8.2.10 page 666 for details.

Collatz–Wielandt formula• Consider positive (or more generally irreducible non-negative matrix) A. For all non-negative non-zero vectors

x let f(x) be the minimum value of [Ax]i / xi taken over all those i such that xi ≠ 0. Then f is a real valuedfunction whose maximum is the Perron–Frobenius eigenvalue r.

First observe that r is attained for x taken to be the Perron-Frobenius eigenvector v. So one needs to prove that valuesf on the other vectors are less or equal. Consider some vector x. Let ξ=f(x), so 0<= ξx <=Ax. Consider w to be theright eigenvector for A. Hence wt ξx <= wt (Ax) = (wt A)x = r wt x . Hence ξ<=r. So proof is finished.One can consult Meyer[1] chapter 8 [2] page 666 for details.

Perron projection as a limit: Ak/rk

Consider positive (or more generally primitive) matrix A, and r its Perron-Frobenius eigenvalue, then:1. There exists a limit Ak/rk for k → ∞, denote it by P.2. P is a projection operator: P2=P, which commutes with A: AP=PA.3. The image of P is one-dimensional and spanned by the Perron-Frobenius eigenvector v (respectively for Pt -

by the Perron-Frobenius eigenvector w for At).4. P= v wt, where v,w are normalized such that w^t v = 1.5. Hence P is a positive operator.

Hence P is a spectral projection for the Perron-Frobenius eigenvalue r, and it is called Perron projection. Payattention: the assertion above is not true for general non-negative irreducible matrices.Actually the claims above (except claim 5) are valid for any matrix M such that there exists an eigenvalue r, which isstrictly greater than the other eigenvalues in absolute value and it is simple root of the characteristic polynomial.(These requirements hold for primitive matrices as it was shown above). The proof of this more general fact issketched below.Proof. One can demonstrate the existence of the limit as follows. Assume M is diagonalizable, so it is conjugate todiagonal matrix with eigenvalues r1, ... , rn on the diagonal (denote r1=r). The matrix Mk/rk will be conjugate (1,(r2/r)k, ... , (rn/r)k), which tends to (1,0,0,...,0), for k → ∞. So the limit exists. The same method works for general M(without assumption M is diagonalizable).The projection and commutativity properties are elementary corollaries of the definition: M Mk/rk= Mk/rk M ; P2 =lim M2k/r2k=P. The third fact is also elementary: M (P u)= M lim Mk/rk u = lim r Mk+1/rk+1 u, so taking the limit onegets M (P u)=r (P u), so image of P lies in the r-eigenspace for M, which is one-dimensional by the assumptions.Denote by v r-eigenvector for M (by w for Mt). Columns of P are multiples of v, because image of P is spanned by it.Respectively rows - of w. So P takes a form (a v wt), for some a. Hence its trace equals to (a wt v). On the other handtrace of projector equals to the dimension of its image. It was proved before that it is not more than one-dimensional.From the definition one sees that P acts identically on the r-eigenvector for M. So it is one-dimensional. So oneconcludes that choosing (wt v)=1, implies P=v wt. The proof is finished.

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Inequalities for Perron–Frobenius eigenvalueFor any non-nonegative matrix A its Perron–Frobenius eigenvalue r satisfies the inequality:

Actually it is not specific to non-negative matrices: for any matrix A and any its eigenvalue λ it is true that. This is immediate corollary of the Gershgorin circle theorem. However there is another proof

which is more direct. Any matrix induced norm satisfies the inequality ||A|| ≥ |λ| for any eigenvalue λ, because ||A|| ≥||Ax||/||x|| = ||λx||/||x|| = |λ|. The infinity norm of a matrix is the maximum of row sums: Hence the desired inequality is exactly ||A||∞ ≥ |λ| applied to non-negative matrix A.Another inequality is:

This fact is specific for non-negative matrices, for general matrices there is nothing similar. To prove it one firstsuppose that A is positive (non just non-negative). Then there exists a positive eigenvector w such that Aw = rw andthe smallest component of w (say wi) is 1. Then r = (Aw)i ≥ the sum of the numbers in row i of A. Thus the minimumrow sum gives a lower bound for r and this observation can be extended to all non-negative matrices by continuity.Another way to argue it is via the Collatz-Wielandt formula. One takes the vector x = (1, 1, ..., 1) and immediatelyobtains the inequality.

Further proofs

Perron projection

The proof now proceeds using spectral decomposition. The trick here is to split the Perron root away from the othereigenvalues. The spectral projection associated with the Perron root is called the Perron projection and it enjoys thefollowing property:

• The Perron projection of an irreducible non-negative square matrix is a positive matrix.Perron's findings and also (1)–(5) of the theorem are corollaries of this result. The key point is that a positiveprojection always has rank one. This means that if A is an irreducible non-negative square matrix then the algebraicand geometric multiplicities of its Perron root are both one. Also if P is its Perron projection then AP = PA = ρ(A)Pso every column of P is a positive right eigenvector of A and every row is a positive left eigenvector. Moreover if Ax= λx then PAx = λPx = ρ(A)Px which means Px = 0 if λ ≠ ρ(A). Thus the only positive eigenvectors are thoseassociated with ρ(A). And there is yet more to come. If A is a primitive matrix with ρ(A) = 1 then it can bedecomposed as P ⊕ (1 − P)A so that An = P + (1 − P)An. As n increases the second of these terms decays to zeroleaving P as the limit of An as n → ∞.The power method is a convenient way to compute the Perron projection of a primitive matrix. If v and w are thepositive row and column vectors that it generates then the Perron projection is just wv/vw. It should be noted that thespectral projections aren't neatly blocked as in the Jordan form. Here they are overlaid on top of one another andeach generally has complex entries extending to all four corners of the square matrix. Nevertheless they retain theirmutual orthogonality which is what facilitates the decomposition.

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Peripheral projection

The analysis when A is irreducible and non-negative is broadly similar. The Perron projection is still positive butthere may now be other eigenvalues of modulus ρ(A) that negate use of the power method and prevent the powers of(1 − P)A decaying as in the primitive case whenever ρ(A) = 1. So enter the peripheral projection which is thespectral projection of A corresponding to all the eigenvalues that have modulus ρ(A) ....

• The peripheral projection of an irreducible non-negative square matrix is a non-negative matrix with a positivediagonal.

Cyclicity

Suppose in addition that ρ(A) = 1 and A has h eigenvalues on the unit circle. If P is the peripheral projection then thematrix R = AP = PA is non-negative and irreducible, Rh = P, and the cyclic group P, R, R2, ...., Rh−1 represents theharmonics of A. The spectral projection of A at the eigenvalue λ on the unit circle is given by the formula

. All of these projections (including the Perron projection) have the same positive diagonal, moreoverchoosing any one of them and then taking the modulus of every entry invariably yields the Perron projection. Somedonkey work is still needed in order to establish the cyclic properties (6)–(8) but it's essentially just a matter ofturning the handle. The spectral decomposition of A is given by A = R ⊕ (1 − P)A so the difference between An andRn is An − Rn = (1 − P)An representing the transients of An which eventually decay to zero. P may be computed as thelimit of Anh as n → ∞.

Caveats

The matrices L = , P = , T = , M = provide simple examples of what

can go wrong if the necessary conditions are not met. It is easily seen that the Perron and peripheral projections of Lare both equal to P, thus when the original matrix is reducible the projections may lose non-negativity and there is nochance of expressing them as limits of its powers. The matrix T is an example of a primitive matrix with zerodiagonal. If the diagonal of an irreducible non-negative square matrix is non-zero then the matrix must be primitivebut this example demonstrates that the converse is false. M is an example of a matrix with several missing spectralteeth. If ω = eiπ/3 then ω6 = 1 and the eigenvalues of M are {1,ω2,ω3,ω4} so ω and ω5 are both absent.

TerminologyA problem that causes confusion is a lack of standardisation in the definitions. For example, some authors use theterms strictly positive and positive to mean > 0 and ≥ 0 respectively. In this article positive means > 0 andnon-negative means ≥ 0. Another vexed area concerns decomposability and reducibility: irreducible is an overloadedterm. For avoidance of doubt a non-zero non-negative square matrix A such that 1 + A is primitive is sometimes saidto be connected. Then irreducible non-negative square matrices and connected matrices are synonymous.[17]

The nonnegative eigenvector is often normalized so that the sum of its components is equal to unity; in this case, theeigenvector is a the vector of a probability distribution and is sometimes called a stochastic eigenvector.Perron–Frobenius eigenvalue and dominant eigenvalue are alternative names for the Perron root. Spectralprojections are also known as spectral projectors and spectral idempotents. The period is sometimes referred to asthe index of imprimitivity or the order of cyclicity.

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Notes[1] Meyer, Carl (2000), Matrix analysis and applied linear algebra (http:/ / www. matrixanalysis. com/ Chapter8. pdf), SIAM,

ISBN 0-89871-454-0,[2] http:/ / www. matrixanalysis. com/ Chapter8. pdf[3] Langville, Amy; Meyer, Carl (2006), Google page rank and beyond (http:/ / pagerankandbeyond. com), Princeton University Press,

ISBN 0-691-12202-4,[4] http:/ / books. google. com/ books?id=hxvB14-I0twC& lpg=PP1& dq=isbn%3A0691122024& pg=PA167#v=onepage& q& f=false[5] Keener, James (1993), "The Perron–Frobenius theorem and the ranking of football teams" (http:/ / links. jstor. org/

sici?sici=0036-1445(199303)35:1<80:TPTATR>2. 0. CO;2-O), SIAM Review (SIAM) 35 (1): 80–93,[6] http:/ / links. jstor. org/ sici?sici=0036-1445%28199303%2935%3A1%3C80%3ATPTATR%3E2. 0. CO%3B2-O[7] Gantmacher, Felix (2000) [1959], The Theory of Matrices, Volume 2 (http:/ / books. google. com/ books?id=cyX32q8ZP5cC& lpg=PA178&

vq=preceding section& pg=PA53#v=onepage& q& f=true), AMS Chelsea Publishing, ISBN 0-8218-2664-6, (1959 edition had different title:"Applications of the theory of matrices". Also the numeration of chapters is different in the two editions.)

[8] http:/ / books. google. com/ books?id=cyX32q8ZP5cC& lpg=PA178& vq=preceding%20section& pg=PA66#v=onepage& q& f=false[9] Kitchens, Bruce (1998), Symbolic dynamics: one-sided, two-sided and countable state markov shifts. (http:/ / books. google. ru/

books?id=mCcdC_5crpoC& lpg=PA195& ots=RbFr1TkSiY& dq=kitchens perron frobenius& pg=PA16#v=onepage& q& f=false), Springer,[10] http:/ / books. google. com/ books?id=cyX32q8ZP5cC& lpg=PA178& vq=preceding%20section& pg=PA62#v=onepage& q& f=true[11] See 2.43 (page 51) in Varga reference below[12] Brualdi, Richard A.; Ryser, Herbert John (1992). Combinatorial Matrix Theory. Cambridge: Cambridge UP. ISBN 0521322650.[13] Brualdi, Richard A.; Cvetkovic, Dragos (2009). A Combinatorial Approach to Matrix Theory and Its Applications. Boca Raton, FL: CRC

Press. ISBN 9781420082234.[14] Mackey, Michael C. (1992). Time's Arrow: The origins of thermodynamic behaviour. New York: Springer-Verlag. ISBN 0387977023.[15] http:/ / books. google. ru/ books?id=cyX32q8ZP5cC& lpg=PR5& dq=Applications%20of%20the%20theory%20of%20matrices&

pg=PA54#v=onepage& q& f=false[16] Smith, Roger (2006), "A Spectral Theoretic Proof of Perron–Frobenius" (ftp:/ / emis. maths. adelaide. edu. au/ pub/ EMIS/ journals/

MPRIA/ 2002/ pa102i1/ pdf/ 102a102. pdf), Mathematical Proceedings of the Royal Irish Academy (The Royal Irish Academy) 102 (1):29–35, doi:10.3318/PRIA.2002.102.1.29,

[17] For surveys of results on irreducibility, see Olga Taussky-Todd and Richard A. Brualdi.

References

Original papers• Perron, Oskar (1907), "Zur Theorie der Matrices", Mathematische Annalen 64 (2): 248–263,

doi:10.1007/BF01449896• Frobenius, Georg (1912), "Ueber Matrizen aus nicht negativen Elementen", Sitzungsber. Königl. Preuss. Akad.

Wiss.: 456–477• Frobenius, Georg (1908), "Über Matrizen aus positiven Elementen, 1", Sitzungsber. Königl. Preuss. Akad. Wiss.:

471–476• Frobenius, Georg (1909), "Über Matrizen aus positiven Elementen, 2", Sitzungsber. Königl. Preuss. Akad. Wiss.:

514–518• Romanovsky, V. (1933), "Sur les zéros des matrices stocastiques" (http:/ / www. numdam. org/

item?id=BSMF_1933__61__213_0), Bulletin de la Société Mathématique de France 61: 213–219• Collatz, Lothar (1942), "Einschließungssatz für die charakteristischen Zahlen von Matrize", Mathematische

Zeitschrift 48 (1): 221–226, doi:10.1007/BF01180013• Wielandt, Helmut (1950), "Unzerlegbare, nicht negative Matrizen", Mathematische Zeitschrift 52 (1): 642–648,

doi:10.1007/BF02230720

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Further reading• Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. ISBN

0-89871-321-8.• Chris Godsil and Gordon Royle, Algebraic Graph Theory, Springer, 2001.• A. Graham, Nonnegative Matrices and Applicable Topics in Linear Algebra, John Wiley&Sons, New York, 1987.• R. A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990• S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability (https:/ / netfiles. uiuc. edu/ meyn/ www/

spm_files/ book. html) London: Springer-Verlag, 1993. ISBN 0-387-19832-6 (2nd edition, Cambridge UniversityPress, 2009)

• Henryk Minc, Nonnegative matrices, John Wiley&Sons, New York, 1988, ISBN 0-471-83966-3• Seneta, E. Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series

in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) ISBN 978-0-387-29765-1• Suprunenko, D.A. (2001), "Perron–Frobenius theorem" (http:/ / eom. springer. de/ P/ p072350. htm), in

Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 (The claim that Aj hasorder n/h at the end of the statement of the theorem is incorrect.)

• Richard S. Varga, Matrix Iterative Analysis, 2nd ed., Springer-Verlag, 2002

Poincaré–Birkhoff–Witt theoremIn the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (Poincaré (1900), G. D. Birkhoff (1937), Witt(1937); frequently contracted to PBW theorem) is a result giving an explicit description of the universal envelopingalgebra of a Lie algebra. The term 'PBW type theorem' or even 'PBW theorem' may also refer to various analogues ofthe original theorem, comparing a filtered algebra to its associated graded algebra, in particular, in the area ofquantum groups.

Statement of the theoremRecall that any vector space V over a field has a basis; this is a set S such that any element of V is a unique (finite)linear combination of elements of S. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases ofwhich the elements are totally ordered by some relation which we denote ≤.If L is a Lie algebra over a field K, there is a canonical K-linear map h from L into the universal enveloping algebraU(L).Theorem. Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finitesequence (x1, x2 ..., xn) of elements of X which is non-decreasing in the order ≤, that is, x1 ≤x2 ≤ ... ≤ xn. Extend h toall canonical monomials as follows: If (x1, x2, ..., xn) is a canonical monomial, let

Then h is injective on the set of canonical monomials and its range is a basis of the K-vector space U(L).Stated somewhat differently, consider Y = h(X). Y is totally ordered by the induced ordering from X. The set ofmonomials

where y1 <y2 < ... < yn are elements of Y, and the exponents are non-negative, together with the multiplicative unit 1,form a basis for U(L). Note that the unit element 1 corresponds to the empty canonical monomial.The multiplicative structure of U(L) is determined by the structure constants in the basis X, that is, the coefficientscu,v,x such that

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This relation allows one to reduce any product of y's to a linear combination of canonical monomials: The structureconstants determine yiyj – yjyi, i.e. what to do in order to change the order of two elements of Y in a product. Thisfact, modulo an inductive argument on the degree of (non-canonical) monomials, shows one can always achieveproducts where the factors are ordered in a non-decreasing fashion.The Poincaré–Birkhoff–Witt theorem can be interpreted as saying that the end result of this reduction is unique anddoes not depend on the order in which one swaps adjacent elements.Corollary. If L is a Lie algebra over a field, the canonical map L → U(L) is injective. In particular, any Lie algebraover a field is isomorphic to a Lie subalgebra of an associative algebra.

More general contextsAlready at its earliest stages, it was known that K could be replaced by any commutative ring, provided that L is afree K-module, i.e., has a basis as above.To extend to the case when L is no longer a free K-module, one needs to make a reformulation that does not usebases. This involves replacing the space of monomials in some basis with the Symmetric algebra, S(L), on L.In the case that K contains the field of rational numbers, one can consider the natural map from S(L) to U(L), sending

a monomial . for , to the element . Then, one has the theorem

that this map is an isomorphism of K-modules.Still more generally and naturally, one can consider U(L) as a filtered algebra, equipped with the filtration given byspecifying that lies in filtered degree . The map of K-modules canonically extendsto a map of algebras, where is the tensor algebra on L (for example, by the universalproperty of tensor algebras), and this is a filtered map equipping with the filtration putting L in degree one(actually, is graded). Then, passing to the associated graded, one gets a canonical morphism

, which kills the elements for , and hence descends to a canonicalmorphism . Then, the (graded) PBW theorem can be reformulated as the statement that, undercertain hypotheses, this final morphism is an isomorphism.This is not true for all K and L (see, for example, the last section of Cohn's 1961 paper), but is true in many cases.These include the aforementioned ones, where either L is a free K-module, or K contains the field of rationalnumbers. More generally, the PBW theorem as formulated above extends to cases such as where (1) L is a flatK-module, (2) L is torsion-free as an abelian group, (3) L is a direct sum of cyclic modules (or all its localizations atprime ideals of K have this property), or (4) K is a Dedekind domain. See, for example, the 1969 paper by Higginsfor these statements.Finally, it is worth noting that, in some of these cases, one also obtains the stronger statement that the canonicalmorphism lifts to a K-module isomorphism , without taking associatedgraded. This is true in the first cases mentioned, where L is a free K-module, or K contains the field of rationalnumbers, using the construction outlined here (in fact, the result is a coalgebra isomorphism, and not merely aK-module isomorphism, equipping both S(L) and U(L) with their natural coalgebra structures such that

for ). This stronger statement, however, might not extend to all of the cases inthe previous paragraph.

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History of the theoremTon-That and Tran have investigated the history of the theorem. They have found out that the majority of the sourcesbefore Bourbaki's 1960 book call it Birkhoff-Witt theorem. Following this old tradition, Fofanova in herencyclopaedic entry says that H. Poincaré obtained the first variant of the theorem. She further says that the theoremwas subsequently completely demonstrated by E. Witt and G.D. Birkhoff. It appears that pre-Bourbaki sources werenot familiar with Poincaré's paper.Birkhoff and Witt do not mention Poincaré's work in their 1937 papers. Cartan and Eilenberg in their 1956 book callthe theorem Poincaré-Witt Theorem and attribute the complete proof to Witt. Bourbaki were the first to use all threenames in their 1960 book. Knapp presents a clear illustration of the shifting tradition. In his 1986 book he calls itBirkhoff-Witt Theorem while in his later 1996 book he switches to Poincaré-Birkhoff-Witt Theorem.It is not clear whether Poincaré's result was complete. Ton-That and Tran conclude that Poincaré had discovered andcompletely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff. On the other hand, theypoint out that Poincaré makes several statements without bothering to prove them. Their own proofs of all the stepsare rather long according to their admission.

References• G.D. Birkhoff, Representability of Lie algebras and Lie groups by matrices Ann. of Math. (2) , 38 : 2 (1937) pp.

526–532• T.S. Fofanova (2001), "Birkhoff–Witt theorem" [1], in Hazewinkel, Michiel, Encyclopaedia of Mathematics,

Springer, ISBN 978-1556080104• P.M. Cohn, A remark on the Birkhoff-Witt theorem, J. London Math. Soc. 38, 197-203, (1963)• P.J. Higgins, Baer Invariants and the Birkhoff-Witt theorem, J. of Alg. 11, 469-482, (1969)• G. Hochschild, The Theory of Lie Groups, Holden-Day, 1965.• H. Poincaré, Sur les groupes continus Trans. Cambr. Philos. Soc. , 18 (1900) pp. 220–225• E. Witt, Treue Darstellung Liescher Ringe [2] J. Reine Angew. Math. , 177 (1937) pp. 152–160• T. Ton-That, T.-D. Tran, Poincaré's proof of the so-called Birkhoff-Witt theorem [3] Rev. Histoire Math., 5

(1999), pp. 249-284.• N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie.

Hermann, Paris, 1960.• A. W. Knapp, Representation theory of semisimple groups. An overview based on examples. Princeton University

Press, 1986.• A. W. Knapp, Lie groups beyond an introduction. Birkhäuser Boston, 1996.

References[1] http:/ / eom. springer. de/ B/ b016540. htm[2] http:/ / gdz. sub. uni-goettingen. de/ dms/ load/ img/ ?PPN=PPN243919689_0177& DMDID=dmdlog17[3] http:/ / smf. emath. fr/ Publications/ RevueHistoireMath/ 5/ pdf/ smf_rhm_5_249-284. pdf

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Polynomial remainder theoremIn algebra, the polynomial remainder theorem or little Bézout's theorem[1] is an application of polynomial longdivision. It states that the remainder of a polynomial divided by a linear divisor is equal to

ExampleLet . Polynomial division of by gives the quotient andthe remainder . Therefore, .

ProofThe polynomial remainder theorem follows from the definition of polynomial long division; denoting the divisor,quotient and remainder by, respectively, , , and , polynomial long division gives a solution of theequation

where the degree of is less than that of .If we take as the divisor, giving the degree of as 0, i.e. :

Setting we obtain:

ApplicationsThe polynomial remainder theorem may be used to evaluate by calculating the remainder, . Althoughpolynomial long division is more difficult than evaluating the function itself, synthetic division is computationallyeasier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remaindertheorem.The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor isa factor. Repeated application of the factor theorem may be used to factorize the polynomial.

References[1] Piotr Rudnicki (2004). "Little Bézout Theorem (Factor Theorem)" (http:/ / mizar. org/ fm/ 2004-12/ pdf12-1/ uproots. pdf). Formalized

Mathematics 12 (1): 49–58. .

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Primitive element theorem 188

Primitive element theoremIn mathematics, more specifically in the area of modern algebra known as field theory, the primitive elementtheorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions thatpossess a primitive element. More specifically, the primitive element theorem characterizes those finite degreeextensions such that there exists with .

TerminologyLet be an arbitrary field extension. An element is said to be a primitive element for when

In this situation, the extension is referred to as a simple extension. Then every element x of E can be writtenin the form

where for all i, and is fixed. That is, if is separable of degree n, there exists such that the set

is a basis for E as a vector space over F.

For instance, the extensions and are simple extensions with primitive elements and x, respectively ( denotes the field of rational functions in the indeterminate x over ).

Existence statementThe interpretation of the theorem changed with the formulation of the theory of Emil Artin, around 1930. From thetime of Galois, the role of primitive elements had been to represent a splitting field as generated by a single element.This (arbitrary) choice of such an element was bypassed in Artin's treatment.[1] At the same time, considerations ofconstruction of such an element receded: the theorem becomes an existence theorem.The following theorem of Artin then takes the place of the classical primitive element theorem.Theorem

Let be a finite degree field extension. Then for some element if and only if thereexist only finitely many intermediate fields K with .A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separabilitywas usually tacitly assumed):Corollary

Let be a finite degree separable extension. Then for some .The corollary applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q hascharacteristic 0 and therefore every extension over Q is separable.

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Primitive element theorem 189

CounterexamplesFor non-separable extensions, necessarily in characteristic p with p a prime number, then at least when the degree[L : K] is p, then L / K has a primitive element, because there are no intermediate subfields. When [L : K] = p2 thenthere may not be a primitive element (and therefore there are infinitely many intermediate fields). This happens, forexample if K is

Fp(T, U),the field of rational functions in two indeterminates T and U over the finite field with p elements, and L is obtainedfrom K by adjoining a p-th root of T, and of U. In fact one can see that for any α in L, the element αp lies in K, but aprimitive element must have degree p2 over K.

Constructive resultsGenerally, the set of all primitive elements for a finite separable extension L / K is the complement of a finitecollection of proper K-subspaces of L, namely the intermediate fields. This statement says nothing for the case offinite fields, for which there is a computational theory dedicated to finding a generator of the multiplicative group ofthe field (a cyclic group), which is a fortiori a primitive element. Where K is infinite, a pigeonhole principle prooftechnique considers the linear subspace generated by two elements and proves that there are only finitely many linearcombinations

with c in K in it, that fail to generate the subfield containing both elements. This is almost immediate as a way ofshowing how Artin's result implies the classical result, and a bound for the number of exceptional c in terms of thenumber of intermediate fields results (this number being something that can be bounded itself by Galois theory and apriori). Therefore in this case trial-and-error is a possible practical method to find primitive elements. See theExample.

ExampleIt is not, for example, immediately obvious that if one adjoins to the field Q of rational numbers roots of bothpolynomials

and

say α and β respectively, to get a field K = Q(α, β) of degree 4 over Q, that the extension is simple and there exists aprimitive element γ in K so that K = Q(γ). One can in fact check that with

the powers γ i for 0 ≤ i ≤ 3 can be written out as linear combinations of 1, α, β and αβ with integer coefficients.Taking these as a system of linear equations, or by factoring, one can solve for α and β over Q(γ) (one gets, forinstance, α= ), which implies that this choice of γ is indeed a primitive element in this example. A simplerargument, assuming the knowledge of all the subfields as given by Galois theory, is to note the independence of1, α, β and αβ over the rationals; this shows that the subfield generated by γ cannot be that generated α or β, nor infact that generated by αβ, exhausting all the subfields of degree 2. Therefore it must be the whole field.

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References• The primitive element theorem at mathreference.com [2]

• The primitive element theorem at planetmath.org [3]

• The primitive element theorem on the site of Cornell's university (pdf file) [4]

Notes[1] Israel Kleiner, A History of Abstract Algebra (2007), p. 64.[2] http:/ / www. mathreference. com/ fld-sep,pet. html[3] http:/ / planetmath. org/ encyclopedia/ ProofOfPrimitiveElementTheorem2. html[4] http:/ / www. math. cornell. edu/ ~kbrown/ 4340/ primitive. pdf

Quillen–Suslin theoremThe Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutativealgebra about the relationship between free modules and projective modules over polynomial rings. It states thatevery finitely generated projective module over a polynomial ring is free.Geometrically, finitely generated projective modules correspond to vector bundles over affine space, and freemodules to trivial vector bundles. Affine space is topologically contractible, so it admits no non-trivial topologicalvector bundles. A simple argument using the exponential exact sequence and the d-bar Poincaré lemma shows that italso admits no non-trivial holomorphic vector bundles. Jean-Pierre Serre, in his 1955 paper "Faisceaux algébriquescohérents", remarked that the equivalent question was not known for algebraic vector bundles: "It is not known ifthere exist projective A-modules of finite type which are not free."[1] Here A is a polynomial ring over a field, that is,A = k[x1, ..., xn].To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as Icould [to the name]."[2] ) The statement is not immediately obvious from the topological and holomorphic cases,because these cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraictrivialization. Instead, the problem turns out to be extremely difficult. Serre made some progress towards a solutionin 1957 when he proved that every finitely generated projective module over a polynomial ring over a field wasstably free, meaning that after forming its direct sum with a finitely generated free module, it became free. Theproblem remained open until 1976, when Daniel Quillen and Andrei Suslin independently proved that the answerwas affirmative. Quillen was awarded the Fields Medal in 1978 in part for his proof of the Serre conjecture. LeonidVaseršteĭn later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's Algebra.

Notes[1] "On ignore s'il existe des A-modules projectifs de type fini qui ne soient pas libres." Serre, FAC, p. 243.[2] Lam, p. 1

References• Serre, Jean-Pierre (March 1955), "Faisceaux algébriques cohérents", Annals of Mathematics. Second Series. 61

(2): 197–278, doi:10.2307/1969915, JSTOR 1969915• Serre, Jean-Pierre (1958), "Modules projectifs et espaces fibrés à fibre vectorielle" (in French), Séminaire P.

Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23• Quillen, Daniel (1976), "Projective modules over polynomial rings", Inventiones Mathematicae 36: 167–171,

doi:10.1007/BF01390008

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QuillenSuslin theorem 191

• Suslin, Andrei A. (1976), "Проективные модули над кольцами многочленов свободны" (in Russian), DokladyAkademii Nauk SSSR 229 (5): 1063–1066. Translated in Soviet Mathematics 17 (4): 1160–1164, 1976.

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York:Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556

An account of this topic is provided by:• Lam, T. Y. (2006), Serre's problem on projective modules, Berlin; New York: Springer Science+Business Media,

pp. 300pp., ISBN 978-3540233176

Rational root theoremIn algebra, the rational root theorem (or rational root test) states a constraint on rational solutions (or roots) of thepolynomial equation

with integer coefficients.If a0 and an are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., thegreatest common divisor of p and q is 1), satisfies• p is an integer factor of the constant term a0, and• q is an integer factor of the leading coefficient an.

Thus, a list of possible rational roots of the equation can be derived using the formula .

The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization ofpolynomials. The integral root theorem is a special case of the rational root theorem if the leadingcoefficient an = 1.

Proofs

An elementary proofLet P(x) = anxn + an-1xn-1 + ... + a1x + a0 for some a0, ..., an ∈ Z, and suppose P(p/q) = 0 for some coprime p, q ∈ Z:

If we shift the constant term to the right hand side and multiply by qn, we get

We see that p times the integer quantity in parentheses equals -a0qn, so p divides a0qn. But p is coprime to q andtherefore to qn, so by (the generalized form of) Euclid's lemma it must divide the remaining factor a0 of the product.If we instead shift the leading term to the right hand side and multiply by qn, we get

And for similar reasons, we can conclude that q divides an.[1]

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Rational root theorem 192

Proof using Gauss's lemmaShould there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatestcommon divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma; this doesnot alter the set of rational roots and only strengthens the divisibility conditions. That lemma says that if thepolynomial factors in ℚ[X], then it also factors in ℤ[X] as a product of primitive polynomials. Now any rational rootp/q corresponds to a factor of degree 1 in ℚ[X] of the polynomial, and its primitive representative is then qx − p,assuming that p and q are coprime. But any multiple in ℤ[X] of qx − p has leading term divisible by q and constantterm divisible by p, which proves the statement. This argument shows that more generally, any irreducible factor ofP can be supposed to have integer coefficients, and leading and constant coefficients dividing the correspondingcoefficients of P.

ExampleFor example, every rational solution of the equation

must be among the numbers symbolically indicated by

± which gives the list of possible answers:

These root candidates can be tested using the Horner scheme (for instance). In this particular case there is exactlyone rational root. If a root candidate does not satisfy the equation, it can be used to shorten the list of remainingcandidates. For example, x = 1 does not satisfy the equation as the left hand side equals 1. This means thatsubstituting x = 1 + t yields a polynomial in t with constant term 1, while the coefficient of t3 remains the same as thecoefficient of x3. Applying the rational root theorem thus yields the following possible roots for t:

Therefore,

Root candidates that do not occur on both lists are ruled out. The list of rational root candidates has thus shrunk tojust x = 2 and x = 2/3.If a root r1 is found, the Horner scheme will also yield a polynomial of degree n − 1 whose roots, together with r1,are exactly the roots of the original polynomial. It may also be the case that none of the candidates is a solution; inthis case the equation has no rational solution. If the equation lacks a constant term a0, then 0 is one of the rationalroots of the equation.

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Rational root theorem 193

Notes[1] D. Arnold, G. Arnold (1993). Four unit mathematics. Edward Arnold. pp. 120–121. ISBN 0340543353.

References• Charles D. Miller, Margaret L. Lial, David I. Schneider: Fundamentals of College Algebra. Scott &

Foresman/Little & Brown Higher Education, 3rd edition 1990, ISBN 0-673-38638-4, pp. 216-221• Phillip S. Jones, Jack D. Bedient: The historical roots of elementary mathematics. Dover Courier Publications

1998, ISBN 0486255638, pp. 116-117 ( online copy (http:/ / books. google. com/ books?id=7xArILpcndYC&pg=PA116) at Google Books)

• Ron Larson: Calculus: An Applied Approach. Cengage Learning 2007, ISBN 9780618958252, pp. 23-24 ( onlinecopy (http:/ / books. google. com/ books?id=bDG7V0OV34C& pg=PA23) at Google Books)

External links• Weisstein, Eric W., " Rational Zero Theorem (http:/ / mathworld. wolfram. com/ RationalZeroTheorem. html)"

from MathWorld.• RationalRootTheorem (http:/ / planetmath. org/ encyclopedia/ RationalRootTheorem. html) at PlanetMath• Another proof that nth roots of integers are irrational, except for perfect nth powers (http:/ / www. cut-the-knot.

org/ Generalization/ RationalRootTheorem. shtml) by Scott E. Brodie• The Rational Roots Test (http:/ / www. purplemath. com/ modules/ rtnlroot. htm) at purplemath.com

Regev's theoremIn algebra, Regev's theorem, proved by Amitai Regev (1971, 1972), states that the tensor product of two PI algebrasis a PI algebra.

References• Regev, Amitai (1971), "Existence of polynomial identities in A⊗FB" [1], Bulletin of the American Mathematical

Society 77 (6): 1067–1069, doi:10.1090/S0002-9904-1971-12869-0, ISSN 0002-9904, MR0284468• Regev, Amitai (1972), "Existence of identities in A⊗B", Israel Journal of Mathematics 11: 131–152,

doi:10.1007/BF02762615, ISSN 0021-2172, MR0314893

References[1] http:/ / projecteuclid. org/ euclid. bams/ 1183533194

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Schreier refinement theorem 194

Schreier refinement theoremIn mathematics, the Schreier refinement theorem of group theory states that any two normal series of subgroups ofa given group have equivalent refinements.The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegantproof of the Jordan–Hölder theorem.

References• Rotman, Joseph (1994). An introduction to the theory of groups. New York: Springer-Verlag.

ISBN 0-387-94285-8.

Schur–Zassenhaus theoremThe Schur–Zassenhaus theorem is a theorem in group theory which states that if is a finite group, and is anormal subgroup whose order is coprime to the order of the quotient group , then is a semidirect productof and .An alternative statement of the theorem is that any normal Hall subgroup of a finite group has a complement in

.It is clear that if we do not impose the coprime condition, the theorem is not true: consider for example the cyclicgroup and its normal subgroup . Then if were a semidirect product of and then would have to contain two elements of order 2, but it only contains one.The Schur–Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classifygroups with a certain set of composition factors?" The other part, which is where the composition factors do not havecoprime orders, is tackled in extension theory.

References• Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. New York: Springer–Verlag.

ISBN 978-0-387-94285-8.• David S. Dummit & Richard M. Foote (2003). Abstract Algebra. Wiley. ISBN 978-0-471-43334-7.• J. S. Milne (2009). Group Theory [1]. Lecture notes.

References[1] http:/ / www. jmilne. org/ math/ CourseNotes/ gt. html

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SerreSwan theorem 195

Serre–Swan theoremIn the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relatesthe geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a commonintuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compactspaces".The two precise formulations of the theorems differ somewhat. The original theorem, as stated by Jean-Pierre Serrein 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closedfield (of any characteristic). The complementary variant stated by Richard Swan in 1962 is more analytic, andconcerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space.

Differential geometrySuppose M is a compact smooth manifold, and a V is a smooth vector bundle over M. The space of smooth sectionsof V is then a module over C∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theoremstates that this module is finitely generated and projective over C∞(M). In other words, every vector bundle is a directsummand of some trivial bundle: M × Cn for some n. The theorem can be proved by constructing a bundleepimorphism from a trivial bundle M × Cn onto V. This can be done by, for instance, exhibiting sections s1...sn withthe property that for each point p, {si(p)} span the fiber over p.The converse is also true: every finitely generated projective module over C∞(M) arises in this way from somesmooth vector bundle on M. Such a module can be viewed as a smooth function f on M with values in the n × nidempotent matrices for some n. The fiber of the corresponding vector bundle over x is then the range of f(x).Therefore, the category of smooth vector bundles on M is equivalent to the category of finitely generated projectivemodules over C∞(M). Details may be found in (Nestruev 2003). This equivalence is extended to the case of anoncompact manifold M (Giachetta et al. 2005).

TopologySuppose X is a compact Hausdorff space, and C(X) is the ring of continuous real-valued functions on X. Analogousto the result above, the category of real vector bundles on X is equivalent to the category of finitely generatedprojective modules over C(X). The same result holds if one replaces "real-valued" by "complex-valued" and "realvector bundle" by "complex vector bundle", but it does not hold if one replace the field by a totally disconnectedfield like the rational numbers.In detail, let Vec(X) be the category of complex vector bundles over X, and let ProjMod(C(X)) be the category offinitely generated projective modules over the C*-algebra C(X). There is a functor Γ : Vec(X)→ProjMod(C(X))which sends each complex vector bundle E over X to the C(X)-module Γ(X,E) of sections. Swan's theorem assertsthat the functor Γ is an equivalence of categories.

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Algebraic geometryThe analogous result in algebraic geometry, due to Serre (1955, §50) applies to vector bundles in the category ofaffine varieties. Let X be an affine variety with structure sheaf OX, and F a coherent sheaf of OX-modules on X. ThenF is the sheaf of germs of a finite-dimensional vector bundle if and only if the space of sections of F, Γ(F,X), is aprojective module over the commutative ring A = Γ(OX,X).

References• Karoubi, Max (1978), K-theory: An introduction, Grundlehren der mathematischen Wissenschaften,

Springer-Verlag, ISBN 978-0387080901• Manoharan, P. (1995), "Generalized Swan's Theorem and its Application", Proceedings of the American

Mathematical Society 123 (10): 3219–3223, doi:10.2307/2160685, JSTOR 2160685.• Serre, Jean-Pierre (1955), "Faisceaux Algebriques Coherents", Annals of Mathematics 61 (2): 197–278,

doi:10.2307/1969915, JSTOR 1969915.• Swan, Richard G. (1962), "Vector Bundles and Projective Modules", Transactions of the American Mathematical

Society 105 (2): 264–277, doi:10.2307/1993627, JSTOR 1993627.• Nestruev, Jet (2003), Smooth manifolds and observables, Graduate texts in mathematics, 220, Springer-Verlag,

ISBN 0-387-95543-7• Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (2005), Geometric and Algebraic Topological Methods in

Quantum Mechanics, World Scientific, ISBN 9812561293.This article incorporates material from Serre-Swan theorem on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

Skolem–Noether theoremIn mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an importantresult in ring theory which characterizes the automorphisms of simple rings.The theorem was first published by Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme(German: On the theory of associative number systems) and later rediscovered by Noether.

Skolem–Noether theoremIn a general formulation, let A and B be simple rings, and let K = Z(B) be the centre of B. Notice that K is a fieldsince given x nonzero in K, the simplicity of B implies that the nonzero two-sided ideal Bx is the whole of B, andhence that x is a unit. Suppose further that the dimension of B over K is finite, i.e. that B is a central simple algebra.Then given K-algebra homomorphisms

f, g : A → Bthere exists a unit b in B such that for all a in A

g(a) = b · f(a) · b−1.In particular, every automorphism of a Brauer algebra is inner.

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References• Thoralf Skolem, Zur Theorie der assoziativen Zahlensysteme, 1927• A proof [1]

References[1] http:/ / www. math. virginia. edu/ ~ww9c/ divalgebras. pdf

Specht's theoremIn mathematics, Specht's theorem gives a necessary and sufficient condition for two matrices to be unitarilyequivalent. It is named after Wilhelm Specht, who proved the theorem in 1940.[1]

Two matrices A and B are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU.[2]

Two matrices which are unitarily equivalent are also similar. Two similar matrices represent the same linear map,but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis toanother orthonormal basis.If A and B are unitarily equivalent, then tr AA* = tr BB*, where tr denotes the trace (in other words, the Frobeniusnorm is a unitary invariant). This follows from the cyclic invariance of the trace: if B = U *AU, then tr BB* = tr U*AUU *A*U = tr AUU *A*UU * = tr AA*, where the second equality is cyclic invariance.[3]

Thus, tr AA* = tr BB* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem givesinfinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses thefollowing definition. A word in two variables, say x and y, is an expression of the form

where m1, n1, m2, n2, …, mp are non-negative integers. The degree of this word is

Specht's theorem: Two matrices A and B are unitarily equivalent if and only if tr W(A, A*) = tr W(B, B*) for allwords W.[4]

The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let n denote the sizeof the matrices A and B. For the case n = 2, the following three conditions are sufficient:[5]

For n = 3, the following seven conditions are sufficient:

[6]

For general n, it suffices to show that tr W(A, A*) = tr W(B, B*) for all words of degree at most

 [7]

It has been conjectured that this can be reduced to an expression linear in n.[8]

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Specht's theorem 198

Notes[1] Specht (1940)[2] Horn & Johnson (1985), Definition 2.2.1[3] Horn & Johnson (1985), Theorem 2.2.2[4] Horn & Johnson (1985), Theorem 2.2.6[5] Horn & Johnson (1985), Theorem 2.2.8[6] Sibirskiǐ (1976), p. 260, quoted by Đoković & Johnson (2007)[7] Pappacena (1997), Theorem 4.3[8] Freedman, Gupta & Guralnick (1997), p. 160

References• Đoković, Dragomir Ž.; Johnson, Charles R. (2007), "Unitarily achievable zero patterns and traces of words in A

and A*", Linear Algebra and its Applications 421 (1): 63–68, doi:10.1016/j.laa.2006.03.002, ISSN 0024-3795.• Freedman, Allen R.; Gupta, Ram Niwas; Guralnick, Robert M. (1997), "Shirshov's theorem and representations of

semigroups" (http:/ / pjm. math. berkeley. edu/ pjm/ 1997/ 181-3/ p07. xhtml), Pacific Journal of Mathematics181 (3): 159–176, doi:10.2140/pjm.1997.181.159, ISSN 0030-8730.

• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press,ISBN 978-0-521-38632-6.

• Pappacena, Christopher J. (1997), "An upper bound for the length of a finite-dimensional algebra", Journal ofAlgebra 197 (2): 535–545, doi:10.1006/jabr.1997.7140, ISSN 0021-8693.

• Sibirskiǐ, K. S. (1976) (in Russian), Algebraic Invariants of Differential Equations and Matrices, Izdat. "Štiinca",Kishinev.

• Specht, Wilhelm (1940), "Zur Theorie der Matrizen. II" (http:/ / gdz. sub. uni-goettingen. de/ dms/ load/ toc/?PPN=PPN37721857X_0050& DMDID=dmdlog6), Jahresbericht der Deutschen Mathematiker-Vereinigung 50:19–23, ISSN 0012-0456.

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Stone's representation theorem for Boolean algebras 199

Stone's representation theorem for BooleanalgebrasIn mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra isisomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra thatemerged in the first half of the 20th century. The theorem was first proved by Stone (1936), and thus named in hishonor. Stone was led to it by his study of the spectral theory of operators on a Hilbert space.

Stone spacesEach Boolean algebra B has an associated topological space, denoted here S(B), called its Stone space. The points inS(B) are the ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean algebra. Thetopology on S(B) is generated by a basis consisting of all sets of the form

where b is an element of B.For any Boolean algebra B, S(B) is a compact totally disconnected Hausdorff space; such spaces are called Stonespaces (also profinite spaces). Conversely, given any topological space X, the collection of subsets of X that areclopen (both closed and open) is a Boolean algebra.

Representation theoremA simple version of Stone's representation theorem states that any Boolean algebra B is isomorphic to the algebraof clopen subsets of its Stone space S(B). The isomorphism sends an element b∈B to the set of all ultrafilters thatcontain b. This is a clopen set because of the choice of topology on S(B) and because B is a Boolean algebra.The full statement of the theorem uses the language of category theory; it states that there is a duality between thecategory of Boolean algebras and the category of Stone spaces. This duality means that in addition to theisomorphisms between Boolean algebras and their Stone spaces, each homomorphism from a Boolean algebra A to aBoolean algebra B corresponds in a natural way to a continuous function from S(B) to S(A). In other words, there is acontravariant functor that gives an equivalence between the categories. This was an early example of a nontrivialduality of categories.The theorem is a special case of Stone duality, a more general framework for dualities between topological spacesand partially ordered sets.The proof requires either the axiom of choice or a weakened form of it. Specifically, the theorem is equivalent to theBoolean prime ideal theorem, a weakened choice principle which states that every Boolean algebra has a prime ideal.

References• Paul Halmos, and Givant, Steven (1998) Logic as Algebra. Dolciani Mathematical Expositions No. 21. The

Mathematical Association of America.• Johnstone, Peter T. (1982) Stone Spaces. Cambridge University Press. ISBN 0-521-23893-5.• Marshall H. Stone (1936) "The Theory of Representations of Boolean Algebras, [1]" Transactions of the American

Mathematical Society 40: 37-111.A monograph available free online:• Burris, Stanley N., and H.P. Sankappanavar, H. P.(1981) A Course in Universal Algebra. [2] Springer-Verlag.

ISBN 3-540-90578-2.

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References[1] http:/ / links. jstor. org/ sici?sici=0002-9947%28193607%2940%3A1%3C37%3ATTORFB%3E2. 0. CO%3B2-8[2] http:/ / www. thoralf. uwaterloo. ca/ htdocs/ ualg. html

Structure theorem for finitely generated modulesover a principal ideal domainIn mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over aprincipal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups androughly states that finitely generated modules can be uniquely decomposed in much the same way that integers havea prime factorization. The result provides a simple framework to understand various canonical form results forsquare matrices over fields.

StatementWhen a vector space over a field F has a finite generating set, then one may extract from it a basis consisting of afinite number n of vectors, and the space is therefore isomorphic to Fn. The corresponding statement with the Fgeneralized to a principal ideal domain R is no longer true, as a finitely generated module over R need not have anybasis. However such a module is still isomorphic to a quotient of some module Rn with n finite (to see this it sufficesto construct the morphism that sends the elements of the canonical basis Rn to the generators of the module, and takethe quotient by its kernel. By changing the choice of generating set, one can in fact describe the module as thequotient of some Rn by a particularly simple submodule, and this is the structure theorem.The structure theorem for finitely generated modules over a principal ideal domain has two statements, which areequivalent by the Chinese remainder theorem:

Invariant factor decompositionEvery finitely generated module M over a principal ideal domain R is isomorphic to a unique one of the form

where and .The elements (up to unit) are a complete set of invariants for finitely generated R-modules, and are calledinvariant factors.The ideals are unique; the elements are unique up to multiplication by a unit, but the order is unique.

Some prefer to separate out the free part and write M as:

where and . The free part is where (in the first formulation) ; these occur at the end, aseverything divides zero.

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Primary decompositionEvery finitely generated module M over a principal ideal domain R is isomorphic to a unique one of the form

where and the are primary ideals. The ideals are unique (up to order); the elements areunique up to multiplication by a unit, and are called the elementary divisors.

Note that in a PID, primary ideals are powers of primes, so .The summands are indecomposable, so the primary decomposition is a decomposition into indecomposablemodules, and thus every finitely generated module over a PID is completely decomposable.Some prefer to separate out the free part (where ) and write M as:

where and the are primary ideals.

ProofsOne proof proceeds as follows:• Every finitely generated module over a PID is also finitely presented because a PID is Noetherian, an even

stronger condition than coherence.• Take a presentation, which is a map (relations to generators), and put it in Smith normal form.This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariantfactors.Another outline of a proof:• Denote by tM the torsion submodule of M. Then M/tM is a finitely generated torsion free module, and such a

module over a commutative PID is a free module of finite rank. As a result, where for a positive integer n.

• For a prime p in R we can then speak of for each prime p. This is asubmodule of tM, and it turns out that each Np is a direct sum of cyclic modules, and that tM is a direct sum of Npfor a finite number of distinct primes p.

• Putting the previous two steps together, M is decomposed into cyclic modules of the indicated types.

CorollariesThis includes the classification of finite-dimensional vector spaces as a special case, where . Since fieldshave no non-trivial ideals, every finitely generated vector space is free.Taking yields the fundamental theorem of finitely generated abelian groups.

Taking classifies linear operators on a finite-dimensional vector space – an operator on a vector spaceis the same as an algebra representation of the polynomial algebra in one variable – where the last invariant factor isthe minimal polynomial, and the product of invariant factors is the characteristic polynomial. Combined with astandard matrix form for , this yields various canonical forms:• invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)• primary decomposition + companion matrix yields primary rational canonical form• primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically

closed field)

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UniquenessWhile the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between M andits canonical form is not unique, and does not even preserve the direct sum decomposition. This follows becausethere are non-trivial automorphisms of these modules which do not preserve the summands.However, one has a canonical torsion submodule T, and similar canonical submodules corresponding to each(distinct) invariant factor, which yield a canonical sequence:

Compare composition series in Jordan–Hölder theorem.

For instance, if , and is one basis, then is another basis, and the

change of basis matrix does not preserve the summand . However, it does preserve the summand,

as this is the torsion submodule (equivalently here, the 2-torsion elements).

Generalizations

GroupsThe Jordan–Hölder theorem is a more general result for finite groups (or modules over an arbitrary ring). In thisgenerality, one obtains a composition series, rather than a direct sum.The Krull–Schmidt theorem and related results give conditions under which a module has something like a primarydecomposition, a decomposition as a direct sum of indecomposable modules in which the summands are unique upto order.

Primary decompositionThe primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and thisresult is called the Lasker–Noether theorem.

Indecomposable modulesBy contrast, unique decomposition into indecomposable submodules does not generalize as far, and the failure ismeasured by the ideal class group, which vanishes for PIDs.For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ringgenerated by two elements. For the ring R = Z[√−5], both the module R and its submodule M generated by 2 and1 + √−5 are indecomposable. While R is not isomorphic to M, R ⊕ R is isomorphic to M ⊕ M; thus the images ofthe M summands give indecomposable submodules L1, L2 < R ⊕ R which give a different decomposition of R ⊕ R.The failure of uniquely factorizing R ⊕ R into a direct sum of indecomposable modules is directly related (via theideal class group) to the failure of the unique factorization of elements of R into irreducible elements of R.

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Non-finitely generated modulesSimilarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the numberof factors may vary. There are Z-submodules A of Q4 which are simultaneously direct sums of two indecomposablemodules and direct sums of three indecomposable modules, showing the analogue of the primary decompositioncannot hold for infinitely generated modules, even over the integers, Z.Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are notfree. For instance, consider the ring Z of integers. A classical example of a torsion-free module which is not free isthe Baer–Specker group, the group of all sequences of integers under termwise addition. In general, the question ofwhich infinitely generated torsion-free abelian groups are free depends on which large cardinals exist. Aconsequence is that any structure theorem for infinitely generated modules depends on a choice of set theory axiomsand may be invalid under a different choice.

References• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press,

ISBN 978-0-201-40751-8• Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: Wiley,

ISBN 978-0-471-43334-7, MR2286236• Hungerford, Thomas W. (1980), Algebra, New York: Springer, pp. 218–226, Section IV.6: Modules over a

Principal Ideal Domain, ISBN 978-0-387-90518-1• Jacobson, Nathan (1985), Basic algebra. I (2 ed.), New York: W. H. Freeman and Company, pp. xviii+499,

ISBN 0-7167-1480-9, MR780184• Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Springer-Verlag,

ISBN 978-0-387-98428-5

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Subgroup testIn Abstract Algebra, the one-step subgroup test is a theorem that states that for any group, a nonempty subset of thatgroup is itself a group if the inverse of any element in the subset multiplied with any other element in the subset isalso in the subset. The two-step subgroup test is a similar theorem which requires the subset to be closed under theoperation and taking of inverses.

One-step subgroup testLet G be a group and let H be a nonempty subset of G. If for all a and b in H, ab-1 is in H, then H is a subgroup of G.

ProofLet G be a group, let H be a nonempty subset of G and assume that for all a and b in H, ab-1 is in H. To prove that His a subgroup of G we must show that H is associative, has an identity, has an inverse for every element and is closedunder the operation. So,• Since the operation of H is the same as the operation of G, the operation is associative since G is a group.• Since H is not empty there exists an element x in H. Then the identity is in H since we can write it as e = x x-1

which is in H by the initial assumption.• Let x be an element of H. Since the identity e is in H it follows that ex-1 = x-1 in H, so the inverse of an element in

H is in H.• Finally, let x and y be elements in H, then since y is in H it follows that y-1 is in H. Hence x(y-1)-1 = xy is in H and

so H is closed under the operation.Thus H is a subgroup of G.

Two-step subgroup testA corollary of this theorem is the two-step subgroup test which states that a nonempty subset of a group is itself agroup if the subset is closed under the operation as well as under the taking of inverses.

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Subring testIn abstract algebra, the subring test is a theorem that states that for any ring, a nonempty subset of that ring is asubring if it is closed under multiplication and subtraction. Note that here that the terms ring and subring are usedwithout requiring a multiplicative identity element.

More formally, let be a ring, and let be a nonempty subset of . If for all one has andfor all one has then is a subring of .If rings are required to have unity, then it must also be assumed that the multiplicative identity is in the subset.

ProofSince is nonempty and closed under subtraction, by the subgroup test it follows that is a group under addition.Hence, is closed under addition, addition is associative, has an additive identity, and every element in hasan additive inverse.Since the operations of are the same as those of it immediately follows that addition is commutative,multiplication is associative, multiplication is left distributive over addition, and multiplication is right distributiveover addition.Thus, is a subring of .

Sylow theoremsIn mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theoremsnamed after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number ofsubgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finitegroup theory and have very important applications in the classification of finite simple groups.For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroupof G, i.e., a subgroup of G which is a p-group (so that the order of any group element is a power of p), and which isnot a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p issometimes written Sylp(G).The Sylow theorems assert a partial converse to Lagrange's theorem that for any finite group G the order (number ofelements) of every subgroup of G divides the order of G. For any prime factor p of the order of a finite group G,there exists a Sylow p-subgroup of G. The order of a Sylow p-subgroup of a finite group G is pn, where n is themultiplicity of p in the order of G, and any subgroup of order pn is a Sylow p-subgroup of G. The Sylow p-subgroupsof a group (for fixed prime p) are conjugate to each other. The number of Sylow p-subgroups of a group for fixedprime p is congruent to 1 mod p.

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Sylow theoremsCollections of subgroups which are each maximal in one sense or another are common in group theory. Thesurprising result here is that in the case of Sylp(G), all members are actually isomorphic to each other and have thelargest possible order: if |G| = pnm with where p does not divide m, then any Sylow p-subgroup P has order|P| = pn. That is, P is a p-group and gcd(|G:P|, p) = 1. These properties can be exploited to further analyze thestructure of G.The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in MathematischeAnnalen.Theorem 1: For any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylowp-subgroup of G, of order pn.The following weaker version of theorem 1 was first proved by Cauchy.Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element oforder p in G .Theorem 2: Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate to each other,i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K.Theorem 3: Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can bewritten as pn · m, where n > 0 and p does not divide m. Let np be the number of Sylow p-subgroups of G. Then thefollowing hold:• np divides m, which is the index of the Sylow p-subgroup in G.• np ≡ 1 mod p.• np = |G : NG(P)|, where P is any Sylow p-subgroup of G and NG denotes the normalizer.

ConsequencesThe Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, pn. Conversely,if a subgroup has order pn, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Dueto the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pn.A very important consequence of Theorem 2 is that the condition np = 1 is equivalent to saying that the Sylowp-subgroup of G is a normal subgroup (there are groups which have normal subgroups but no normal Sylowsubgroups, such as S4).

Sylow theorems for infinite groupsThere is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in an infinite groupto be a p-subgroup (that is, every element in it has p-power order) which is maximal for inclusion among allp-subgroups in the group. Such subgroups exist by Zorn's lemma.Theorem: If K is a Sylow p-subgroup of G, and np = |Cl(K)| is finite, then every Sylow p-subgroup is conjugate to K,and np ≡ 1 mod p, where Cl(K) denotes the conjugacy class of K.

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Examples

In all reflections are conjugate, asreflections correspond to Sylow 2-subgroups.

In reflections no longer correspond toSylow 2-subgroups, and fall into two conjugacy

classes.

A simple illustration of Sylow subgroups and the Sylow theorems arethe dihedral group of the n-gon, For n odd, is thehigher power of 2 dividing the order, and thus subgroups of order 2 areSylow subgroups. These are the groups generated by a reflection, ofwhich there are n, and they are all conjugate under rotations;geometrically the axes of symmetry pass through a vertex and a side.By contrast, if n is even, then 4 divides the order of the group, andthese are no longer Sylow subgroups, and in fact they fall into twoconjugacy classes, geometrically according to whether they passthrough two vertices or two faces. These are related by an outerautomorphism, which can be represented by rotation through half the minimal rotation in the dihedral group.

Example applications

Cyclic group ordersSome numbers n are such that every group of order n is cyclic. One can show that n = 15 is such a number using theSylow theorems: Let G be a group of order 15 = 3 · 5 and n3 be the number of Sylow 3-subgroups. Then and

. The only value satisfying these constraints is 1; therefore, there is only one subgroup oforder 3, and it must be normal (since it has no distinct conjugates). Similarly, n5 must divide 3, and n5 must equal 1(mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection ofthese two subgroups is trivial, and so G must be the internal direct product of groups of order 3 and 5, that is thecyclic group of order 15. Thus, there is only one group of order 15 (up to isomorphism).

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Small groups are not simpleA more complex example involves the order of the smallest simple group which is not cyclic. Burnside's paqb

theorem states that if the order of a group is the product of two prime powers, then it is solvable, and so the group isnot simple, or is of prime order and is cyclic. This rules out every group up to order 30 (= 2 · 3 · 5).If G is simple, and |G| = 30, then n3 must divide 10 ( = 2 · 5), and n3 must equal 1 (mod 3). Therefore n3 = 10, sinceneither 4 nor 7 divides 10, and if n3 = 1 then, as above, G would have a normal subgroup of order 3, and could not besimple. G then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus theidentity). This means G has at least 20 distinct elements of order 3. As well, n5 = 6, since n5 must divide 6 ( = 2 · 3),and n5 must equal 1 (mod 5). So G also has 24 distinct elements of order 5. But the order of G is only 30, so a simplegroup of order 30 cannot exist.Next, suppose |G| = 42 = 2 · 3 · 7. Here n7 must divide 6 ( = 2 · 3) and n7 must equal 1 (mod 7), so n7 = 1. So, asbefore, G can not be simple.On the other hand for |G| = 60 = 22 · 3 · 5, then n3 = 10 and n5 = 6 is perfectly possible. And in fact, the smallestsimple non-cyclic group is A5, the alternating group over 5 elements. It has order 60, and has 24 cyclic permutationsof order 5, and 20 of order 3.

Fusion resultsFrattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. Aslight generalization known as Burnside's fusion theorem states that if G is a finite group with Sylow p-subgroup Pand two subsets A and B normalized by P, then A and B are G-conjugate if and only if they are NG(P)-conjugate. Theproof is a simple application of Sylow's theorem: If B=Ag, then the normalizer of B contains not only P but also Pg

(since Pg is contained in the normalizer of Ag). By Sylow's theorem P and Pg are conjugate not only in G, but in thenormalizer of B. Hence gh−1 normalizes P for some h that normalizes B, and then Agh−1 = Bh−1 = B, so that A and Bare NG(P)-conjugate. Burnside's fusion theorem can be used to give a more power factorization called a semidirectproduct: if G is a finite group whose Sylow p-subgroup P is contained in the center of its normalizer, then G has anormal subgroup K of order coprime to P, G = PK and P∩K = 1, that is, G is p-nilpotent.Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control aSylow p-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited atseveral stages of the classification of finite simple groups, and for instance defines the case divisions used in theAlperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedralgroup. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sortsof elements are used in the conjugation.

Proof of the Sylow theoremsThe Sylow theorems have been proved in a number of ways, and the history of the proofs themselves are the subjectof many papers including (Waterhouse 1980), (Scharlau 1988), (Casadio & Zappa 1990), (Gow 1994), and to someextent (Meo 2004).One proof of the Sylow theorems exploit the notion of group action in various creative ways. The group G acts onitself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of theSylow theorems. The following proofs are based on combinatorial arguments of (Wielandt 1959). In the following,we use a | b as notation for "a divides b" and a b for the negation of this statement.

Theorem 1: A finite group G whose order |G| is divisible by a prime power pk has a subgroup of orderpk.

Proof: Let |G| = pkm = pk+ru such that p does not divide u, and let Ω denote the set of subsets of G of size pk. G acts on Ω by left multiplication. The orbits Gω = {gω | g ∈ G} of the ω ∈ Ω are the equivalence classes under the action

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of G.For any ω ∈ Ω consider its stabilizer subgroup Gω. For any fixed element α ∈ ω the function [g ↦ gα] maps Gω to ωinjectively: for any two g,h ∈ Gω we have that gα = hα implies g = h, because α ∈ ω ⊆ G means that one may cancelon the right. Therefore pk = |ω| ≥ |Gω|.On the other hand

and no power of p remains in any of the factors inside the product on the right. Hence νp(|Ω|) = νp(m) = r. Let R ⊆ Ωbe a complete representation of all the equivalence classes under the action of G. Then,

Thus, there exists an element ω ∈ R such that s := νp(|Gω|) ≤ νp(|Ω|) = r. Hence |Gω| = psv where p does not divide v.By the stabilizer-orbit-theorem we have |Gω| = |G| / |Gω| = pk+r-su / v. Therefore pk | |Gω|, so pk ≤ |Gω| and Gω isthe desired subgroup.

Lemma: Let G be a finite p-group, let G act on a finite set Ω, and let Ω0 denote the set of points of Ωthat are fixed under the action of G. Then |Ω| ≡ |Ω0| mod p.

Proof: Write Ω as a disjoint sum of its orbits under G. Any element x ∈ Ω not fixed by G will lie in an orbit of order|G|/|Gx| (where Gx denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately.

Theorem 2: If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element gin G such that g−1Hg ≤ P. In particular, all Sylow p-subgroups of G are conjugate to each other (andtherefore isomorphic), i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in Gwith g−1Hg = K.

Proof: Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H onΩ, we see that |Ω0| ≡ |Ω| = [G : P] mod p. Now p [G : P] by definition so p |Ω0|, hence in particular |Ω0| ≠ 0 sothere exists some gP ∈ Ω0. It follows that for some g ∈ G and ∀ h ∈ H we have hgP = gP so g−1hgP ⊆ P andtherefore g−1Hg ≤ P. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg−1| so that H = gPg−1 for some g ∈ G.

Theorem 3: Let q denote the order of any Sylow p-subgroup of a finite group G. Then np | |G|/q and np≡ 1 mod p.

Proof: By Theorem 2, np = [G : NG(P)], where P is any such subgroup, and NG(P) denotes the normalizer of P in G,so this number is a divisor of |G|/q. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω byconjugation. Let Q ∈ Ω0 and observe that then Q = xQx−1 for all x ∈ P so that P ≤ NG(Q). By Theorem 2, P and Qare conjugate in NG(Q) in particular, and Q is normal in NG(Q), so then P = Q. It follows that Ω0 = {P} so that, bythe Lemma, |Ω| ≡ |Ω0| = 1 mod p.

AlgorithmsThe problem of finding a Sylow subgroup of a given group is an important problem in computational group theory.One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H] isdivisible by p, then the normalizer N = NG(H) of H in G is also such that [N:H] is divisible by p. In other words, apolycyclic generating system of a Sylow p-subgroup can be found by starting from any p-subgroup H (including theidentity) and taking elements of p-power order contained in the normalizer of H but not in H itself. The algorithmicversion of this (and many improvements) is described in textbook form in (Butler 1991, Chapter 16), including thealgorithm described in (Cannon 1971). These versions are still used in the GAP computer algebra system.In permutation groups, it has been proven in (Kantor 1985a, 1985b, 1990; Kantor & Taylor 1988) that a Sylow p-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the

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number of generators). These algorithms are described in textbook form in (Seress 2003), and are now becomingpractical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of thisalgorithm are used in the Magma computer algebra system.

Notes

References• Sylow, L. (1872), "Théorèmes sur les groupes de substitutions" (http:/ / resolver. sub. uni-goettingen. de/

purl?GDZPPN002242052) (in French), Math. Ann. 5 (4): 584–594, doi:10.1007/BF01442913, JFM 04.0056.02

Proofs• Casadio, Giuseppina; Zappa, Guido (1990), "History of the Sylow theorem and its proofs" (in Italian), Boll. Storia

Sci. Mat. 10 (1): 29–75, ISSN 0392-4432, MR1096350, Zbl 0721.01008• Gow, Rod (1994), "Sylow's proof of Sylow's theorem", Irish Math. Soc. Bull. (33): 55–63, ISSN 0791-5578,

MR1313412, Zbl 0829.01011• Kammüller, Florian; Paulson, Lawrence C. (1999), "A formal proof of Sylow's theorem. An experiment in

abstract algebra with Isabelle HOL" (http:/ / www. cl. cam. ac. uk/ users/ lcp/ papers/ Kammueller/ sylow. pdf), J.Automat. Reason. 23 (3): 235–264, doi:10.1023/A:1006269330992, ISSN 0168-7433, MR1721912,Zbl 0943.68149

• Meo, M. (2004), "The mathematical life of Cauchy's group theorem", Historia Math. 31 (2): 196–221,doi:10.1016/S0315-0860(03)00003-X, ISSN 0315-0860, MR2055642, Zbl 1065.01009

• Scharlau, Winfried (1988), "Die Entdeckung der Sylow-Sätze" (in German), Historia Math. 15 (1): 40–52,doi:10.1016/0315-0860(88)90048-1, ISSN 0315-0860, MR931678, Zbl 0637.01006

• Waterhouse, William C. (1980), "The early proofs of Sylow's theorem", Arch. Hist. Exact Sci. 21 (3): 279–290,doi:10.1007/BF00327877, ISSN 0003-9519, MR575718, Zbl 0436.01006

• Wielandt, Helmut (1959), "Ein Beweis für die Existenz der Sylowgruppen" (in German), Arch. Math. 10 (1):401–402, doi:10.1007/BF01240818, ISSN 0003-9268, MR0147529, Zbl 0092.02403

Algorithms• Butler, G. (1991), Fundamental Algorithms for Permutation Groups, Lecture Notes in Computer Science, 559,

Berlin, New York: Springer-Verlag, doi:10.1007/3-540-54955-2, ISBN 978-3-540-54955-0, MR1225579,Zbl 0785.20001

• Cannon, John J. (1971), "Computing local structure of large finite groups", Computers in Algebra and NumberTheory (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970), SIAM-AMS Proc., 4, Providence, RI: AMS,pp. 161–176, ISSN 0160-7634, MR0367027, Zbl 0253.20027

• Kantor, William M. (1985a), "Polynomial-time algorithms for finding elements of prime order and Sylowsubgroups", J. Algorithms 6 (4): 478–514, doi:10.1016/0196-6774(85)90029-X, ISSN 0196-6774, MR813589,Zbl 0604.20001

• Kantor, William M. (1985b), "Sylow's theorem in polynomial time", J. Comput. System Sci. 30 (3): 359–394,doi:10.1016/0022-0000(85)90052-2, ISSN 1090-2724, MR805654, Zbl 0573.20022

• Kantor, William M.; Taylor, Donald E. (1988), "Polynomial-time versions of Sylow's theorem", J. Algorithms 9(1): 1–17, doi:10.1016/0196-6774(88)90002-8, ISSN 0196-6774, MR925595, Zbl 0642.20019

• Kantor, William M. (1990), "Finding Sylow normalizers in polynomial time", J. Algorithms 11 (4): 523–563,doi:10.1016/0196-6774(90)90009-4, ISSN 0196-6774, MR1079450, Zbl 0731.20005

• Seress, Ákos (2003), Permutation Group Algorithms, Cambridge Tracts in Mathematics, 152, CambridgeUniversity Press, ISBN 978-0-521-66103-4, MR1970241, Zbl 1028.20002

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Sylvester's determinant theoremIn matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants.It is named after James Joseph Sylvester.The theorem states that if A, B are matrices of size p × n and n × p respectively, then

where Ia is the identity matrix of order a.[1]

It is closely related to the Matrix determinant lemma and its generalization.This theorem is useful in developing a Bayes estimator for multivariate Gaussian distributions.Sylvester (1857) stated this theorem without proof.

External links[Related post [2]] on the blog of Terence Tao.

References[1] David A. Harville. Matrix Algebra From a Statistician's Perspective. Springer, 2008, Pages 416[2] https:/ / terrytao. wordpress. com/ 2010/ 12/ 17/ the-mesoscopic-structure-of-gue-eigenvalues/

Sylvester's law of inertiaSylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a realquadratic form that remain invariant under a change of coordinates. Namely, if A is the symmetric matrix thatdefines the quadratic form, and S is any invertible matrix such that D = SAST is diagonal, then the number ofnegative elements in the diagonal of D is always the same, for all such S; and the same goes for the number ofpositive elements.This property is named after J. J. Sylvester who published its proof in 1852.[1] [2]

Statement of the theoremLet A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size is said totransform A into another symmetric matrix B = SAST, also of order n, where ST is the transpose of S. If A is thecoefficient matrix of some quadratic form of Rn, then B is the matrix for the same form after the change ofcoordinates defined by S.A symmetric matrix A can always be transformed in this way into a diagonal matrix D which has only entries 0, +1and −1 along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is aninvariant of A, i.e. it does not depend on the matrix S used.The number of +1s, denoted n+, is called the positive index of inertia of A, and the number of −1s, denoted n−, iscalled the negative index of inertia. The number of 0s, denoted n0, is the dimension of the kernel of A, and also thecorank of A. These numbers satisfy an obvious relation

The difference sign(A) = n− − n+) is usually called the signature of A. (However, some authors use that term for thewhole triple (n0, n+, n−) consisting of the corank and the positive and negative indices of inertia of A.)

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Sylvester's law of inertia 212

If the matrix A has the property that every principal upper left k×k minor Δk is non-zero then the negative index ofinertia is equal to the number of sign changes in the sequence

Statement in terms of eigenvaluesThe positive and negative indices of a symmetric matrix A are also the number of positive and negative eigenvaluesof A. Any symmetric real matrix A has an eigendecomposition of the form QEQT where E is a diagonal matrixcontaining the eigenvalues of A, and Q is an orthonormal square matrix containing the eigenvectors. The matrix Ecan be written E = WDWT where D is diagonal with entries 0, +1, or −1, and W is diagonal with Wii = √|Eii|. Thematrix S = QW transforms D to A.

Law of inertia for quadratic formsIn the context of quadratic forms, a real quadratic form Q in n variables (or on an n-dimensional real vector space)can by a suitable change of basis be brought to the diagonal form

with each ai ∈ {0, 1, −1}. Sylvester's law of inertia states that the number of coefficients of a given sign is aninvariant of Q, i.e. does not depend on a particular choice of diagonalizing basis. Expressed geometrically, the law ofinertia says that all maximal subspaces on which the restriction of the quadratic form is positive definite(respectively, negative definite) have the same dimension. These dimensions are the positive and negative indices ofinertia.

References[1] Sylvester, J J (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal

substitutions to the form of a sum of positive and negative squares" (http:/ / www. maths. ed. ac. uk/ ~aar/ sylv/ inertia. pdf). PhilosophicalMagazine (Ser. 4) 4 (23): 138–142. doi:10.1080/14786445208647087. . Retrieved 2008-06-27.

[2] Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 0-19-853248-2.

External links• Sylvester's law (http:/ / planetmath. org/ encyclopedia/ SylvestersLaw. html) on PlanetMath.

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Takagi existence theorem 213

Takagi existence theoremIn class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusionreversing correspondence between the finite abelian extensions of K (in a fixed algebraic closure of K) and thegeneralized ideal class groups defined via a modulus of K.It is called an existence theorem because a main burden of the proof is to show the existence of enough abelianextensions of K.

FormulationHere a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K withpositive integer exponents. The archimedean valuations that might appear in a modulus include only those whosecompletions are the real numbers (not the complex numbers); they may be identified with orderings on K and occuronly to exponent one.The modulus m is a product of a non-archimedean (finite) part mf and an archimedean (infinite) part m∞. Thenon-archimedean part mf is a nonzero ideal in the ring of integers OK of K and the archimedean part m∞ is simply aset of real embeddings of K. Associated to such a modulus m are two groups of fractional ideals. The larger one, Im,is the group of all fractional ideals relatively prime to m (which means these fractional ideals do not involve anyprime ideal appearing in mf). The smaller one, Pm, is the group of principal fractional ideals (u/v) where u and v arenonzero elements of OK which are prime to mf, u ≡ v mod mf, and u/v > 0 in each of the orderings of m∞. (It isimportant here that in Pm, all we require is that some generator of the ideal has the indicated form. If one does, othersmight not. For instance, taking K to be the rational numbers, the ideal (3) lies in P4 because (3) = (−3) and −3 fits thenecessary conditions. But (3) is not in P4∞ since here it is required that the positive generator of the ideal is 1 mod 4,which is not so.) For any group H lying between Im and Pm, the quotient Im/H is called a generalized ideal classgroup.It is these generalized ideal class groups which correspond to abelian extensions of K by the existence theorem, andin fact are the Galois groups of these extensions. That generalized ideal class groups are finite is proved along thesame lines of the proof that the usual ideal class group is finite, well in advance of knowing these are Galois groupsof finite abelian extensions of the number field.

A well-defined correspondenceStrictly speaking, the correspondence between finite abelian extensions of K and generalized ideal class groups is notquite one-to-one. Generalized ideal class groups defined relative to different moduli can give rise to the same abelianextension of K, and this is codified a priori in a somewhat complicated equivalence relation on generalized idealclass groups.In concrete terms, for abelian extensions L of the rational numbers, this corresponds to the fact that an abelianextension of the rationals lying in one cyclotomic field also lies in infinitely many other cyclotomic fields, and foreach such cyclotomic overfield one obtains by Galois theory a subgroup of the Galois group corresponding to thesame field L.In the idelic formulation of class field theory, one obtains a precise one-to-one correspondence between abelianextensions and appropriate groups of ideles, where equivalent generalized ideal class groups in the ideal-theoreticlanguage correspond to the same group of ideles.

Page 219: Theorems in Algebra

Takagi existence theorem 214

Earlier workA special case of the existence theorem is when m = 1 and H = P1. In this case the generalized ideal class group isthe ideal class group of K, and the existence theorem says there exists a unique abelian extension L/K with Galoisgroup isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is calledthe Hilbert class field. It was conjectured by David Hilbert to exist, and existence in this special case was proved byFurtwängler in 1907, before Takagi's general existence theorem.A further and special property of the Hilbert class field, not true of other abelian extensions of a number field, is thatall ideals in a number field become principal in the Hilbert class field. It required Artin and Furtwängler to prove thatprincipalization occurs.

HistoryThe existence theorem is due to Takagi, who proved it in Japan during the isolated years of World War I. Hepresented it at the International Congress of Mathematicians in 1920, leading to the development of the classicaltheory of class field theory during the 1920s. At Hilbert's request, the paper was published in MathematischeAnnalen in 1925.

References• Helmut Hasse, History of Class Field Theory, pp. 266–279 in Algebraic Number Theory, eds. J. W. S. Cassels

and A. Fröhlich, Academic Press 1967. (See also the rich bibliography attached to Hasse's article.)

Three subgroups lemmaIn mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. Itis a consequence of the Hall–Witt identity.

NotationIn that which follows, the following notation will be employed:• If H and K are subgroups of a group G, the commutator of H and K will be denoted by [H,K]; if L is a third

subgroup, the convention that [H,K,L] = [[H,K],L] will be followed.• If x and y are elements of a group G, the conjugate of x by y will be denoted by .• If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).

StatementLet X, Y and Z be subgroups of a group G, and assume

and Then .[1]

More generally, if , then if and , then .[2]

Proof and the Hall–Witt identityHall–Witt identity

If , then

Page 220: Theorems in Algebra

Three subgroups lemma 215

Proof of the Three subgroups lemma

Let , , and . Then , and by the Hall–Witt identity above,it follows that and so . Therefore, for all and

. Since these elements generate , we conclude that and hence.

Notes[1] Isaacs, Lemma 8.27, p. 111[2] Isaacs, Corollary 8.28, p. 111

References• I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company.

ISBN 0-534-19002-2.

Trichotomy theoremIn mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type andrank at least 3 into three classes. It was proved by Aschbacher (1981, 1983) for rank 3 and by Gorenstein & Lyons(1983) for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groupsof "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), andgroups of uniqueness type, where Aschbacher proved that there are no simple groups.

References• Aschbacher, Michael (1981), "Finite groups of rank 3. I", Inventiones Mathematicae 63 (3): 357–402,

doi:10.1007/BF01389061, ISSN 0020-9910, MR620676• Aschbacher, Michael (1983), "Finite groups of rank 3. II", Inventiones Mathematicae 71 (1): 51–163,

doi:10.1007/BF01393339, ISSN 0020-9910, MR688262• Gorenstein, D.; Lyons, Richard (1983), "The local structure of finite groups of characteristic 2 type" [1], Memoirs

of the American Mathematical Society 42 (276): vii+731, ISBN 978-0-8218-2276-0, ISSN 0065-9266,MR690900

References[1] http:/ / books. google. com/ books?isbn=978-0821822760

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Walter theorem 216

Walter theoremIn mathematics, the Walter theorem, proved by Walter (1967, 1969), describes the finite groups whose Sylow2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler proof.

StatementWalter's theorem states that if G is a finite group whose 2-sylow subgroups are abelian, then G/O(G) has a normalsubgroup of odd index that is a product of groups each of which is a 2-group of one of the simple groups PSL2(q) forq = 2n or q = 3 or 5 mod 8, or the Janko group J1, or Ree groups 2G2(32n+1).The original statement of Walter's theorem did not quite identify the Ree groups, but only stated that thecorresponding groups have a similar subgroup structure as Ree groups. Thompson (1967, 1972, 1977) and Bombieri,Odlyzko & Hunt (1980) later showed that they are all Ree groups, and Enguehard (1986) gave a unified expositionof this result.

References• Bender, Helmut (1970), "On groups with abelian Sylow 2-subgroups", Mathematische Zeitschrift 117: 164–176,

doi:10.1007/BF01109839, ISSN 0025-5874, MR0288180• Bombieri, Enrico; Odlyzko, Andrew; Hunt, D. (1980), "Thompson's problem (σ2=3)", Inventiones Mathematicae

58 (1): 77–100, doi:10.1007/BF01402275, ISSN 0020-9910, MR570875• Enguehard, Michel (1986), "Caractérisation des groupes de Ree", Astérisque (142): 49–139, ISSN 0303-1179,

MR873958• Thompson, John G. (1967), "Toward a characterization of E2

*(q)", Journal of Algebra 7: 406–414,doi:10.1016/0021-8693(67)90080-4, ISSN 0021-8693, MR0223448

• Thompson, John G. (1972), "Toward a characterization of E2*(q). II", Journal of Algebra 20: 610–621,

doi:10.1016/0021-8693(72)90074-9, ISSN 0021-8693, MR0313377• Thompson, John G. (1977), "Toward a characterization ofE2

*(q). III", Journal of Algebra 49 (1): 162–166,doi:10.1016/0021-8693(77)90276-9, ISSN 0021-8693, MR0453858

• Walter, John H. (1967), "Finite groups with abelian Sylow 2-subgroups of order 8", Inventiones Mathematicae 2:332–376, doi:10.1007/BF01428899, ISSN 0020-9910, MR0218445

• Walter, John H. (1969), "The characterization of finite groups with abelian Sylow 2-subgroups.", Annals ofMathematics. Second Series 89: 405–514, ISSN 0003-486X, JSTOR 1970648, MR0249504

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Wedderburn's little theorem 217

Wedderburn's little theoremIn mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finiterings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings.

HistoryThe original proof was given by Joseph Wedderburn in 1905, who went on to prove it two other ways. Another proofwas given by Leonard Eugene Dickson shortly after Wedderburn's original proof, and Dickson acknowledgedWedderburn's priority. However, as noted in (Parshall 1983), Wedderburn's first proof was incorrect – it had a gap –and his subsequent proofs came after he had read Dickson's correct proof. On this basis, Parshall argues that Dicksonshould be credited with the first correct proof.A simplified version of the proof was later given by Ernst Witt. Witt's proof is sketched below. Alternatively, thetheorem is a consequence of the Skolem–Noether theorem.

Sketch of proofLet be a finite domain. For each nonzero , the map

is injective; thus, surjective. Hence, has a left inverse. By the same argument, has a right inverse. A is thus askew-field. Since the center of is a field, is a vector space over with finite dimension n. Ourobjective is then to show . If is the order of , then A has order . For each that is not inthe center, the centralizer of x has order where d divides n. Viewing , and as groupsunder multiplication, we can write the class equation

where the sum is taken over all representatives that is not in and d are the numbers discussed above.and both admit factorization in terms of cyclotomic polynomials . After cancellation, we see

that divides and , so it must divide . So we reach contradiction unless .

References• Parshall, K. H. (1983), In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn,

Leonard Dickson, and Oswald Veblen, Archives of International History of Science, 33, pp. 274–99

External links• Proof of Wedderburn's Theorem at Planet Math [1]

References[1] http:/ / planetmath. org/ ?op=getobj& from=objects& id=3627

Page 223: Theorems in Algebra

Weil conjecture on Tamagawa numbers 218

Weil conjecture on Tamagawa numbersIn mathematics, the Weil conjecture on Tamagawa numbers is a result about algebraic groups formulated byAndré Weil in the late 1950s and proved in 1989. It states that the Tamagawa number τ(G) is 1 for any simplyconnected semisimple algebraic group G defined over a number field K.Here simply connected is in the algebraic group theory sense of not having a proper algebraic covering, which is notalways the topologists' meaning.

HistoryWeil checked this in enough classical group cases to propose the conjecture. In particular for spin groups it impliesthe known Smith–Minkowski–Siegel mass formula.Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. J. G. M. Mars gavefurther results during the 1960s.K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. Kottwitz (1988) proved it for allgroups satisfying the Hasse principle, which at the time was known for all groups without E8 factors. V. I.Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strongapproximation in algebraic groups), thus completing the proof of Weil's conjecture.

References• Hazewinkel, Michiel, ed. (2001), "Tamagawa number" [1], Encyclopaedia of Mathematics, Springer,

ISBN 978-1556080104• Chernousov, V. I. (1989), "The Hasse principle for groups of type E8", Soviet Math. Dokl. 39: 592–596,

MR1014762• Kottwitz, Robert E. (1988), "Tamagawa numbers", Ann. Of Math. (2) (Annals of Mathematics) 127 (3): 629–646,

doi:10.2307/2007007, JSTOR 2007007, MR0942522.• Lai, K. F. (1980), "Tamagawa number of reductive algebraic groups" [2], Compositio Mathematica 41 (2):

153–188, MR581580• Langlands, R. P. (1966), "The volume of the fundamental domain for some arithmetical subgroups of Chevalley

groups", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Providence, R.I.: Amer.Math. Soc., pp. 143–148, MR0213362

• Voskresenskii, V. E. (1991), Algebraic Groups and their Birational Invariants, AMS translation

References[1] http:/ / eom. springer. de/ T/ t092060. htm[2] http:/ / www. numdam. org/ item?id=CM_1980__41_2_153_0

Page 224: Theorems in Algebra

Witt's theorem 219

Witt's theorem"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of ordertheory.

Witt theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometrybetween two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the wholespace. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms overarbitrary fields. The theorem applies to classification of quadratic forms over k and in particular allows one to definethe Witt group W(k) which controls the "stable" theory of quadratic forms over the field k.

Statement of the theoremLet (V, b) be a finite-dimensional vector space over an arbitrary field k together with a nondegenerate symmetric orskew-symmetric bilinear form. If f: U→U' is an isometry between two subspaces of V then f extends to an isometryof V.Witt's theorem implies that the dimension of a maximal isotropic subspace of V is an invariant, called the index orWitt index of b, and moreover, that the isometry group of (V, b) acts transitively on the set of maximal isotropicsubspaces. This fact plays an important role in the structure theory and representation theory of the isometry groupand in the theory of reductive dual pairs.

Witt's cancellation theoremLet (V, q), (V1, q1), (V2, q2) be three quadratic spaces over a field k. Assume that

Then the quadratic spaces (V1, q1) and (V2, q2) are isometric:

In other words, the direct summand (V, q) appearing in both sides of an isomorphism between quadratic spaces maybe "cancelled".

Witt's decomposition theoremLet (V, q) be a quadratic space over a field k. Then it admits a Witt decomposition:

where V0=ker q is the radical of q, (Va, qa) is an anisotropic quadratic space and (Vh, qh) is a hyperbolic quadraticspace. Moreover, the anisotropic summand and the hyperbolic summand in a Witt decomposition of (V, q) aredetermined uniquely up to isomorphism.

References• O. Timothy O'Meara, Introduction to Quadratic Forms, Springer-Verlag, 1973

Page 225: Theorems in Algebra

Z* theorem 220

Z* theoremIn mathematics, George Glauberman's Z* theorem states that if G is a finite group and T is a Sylow 2-subgroup of Gcontaining an involution not conjugate in G to any other element of T, then the involution lies in Z*(G). Thesubgroup Z*(G) is the inverse image in G of the center of G/O(G), where O(G) is the maximal normal subgroup of Gof odd order.This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer-Suzuki theorem to deal with some smallcases).The original paper (Glauberman 1966) gave several criteria for an element to lie outside Z*(G). Its theorem 4 states:For an element t in T, it is necessary and sufficient for t to lie outside Z*(G) that there is some g in G and abeliansubgroup U of T satisfying the following properties:1. g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U)∩NG(CT(U))2. t is contained in U and tg ≠ gt3. U is generated by the N-conjugates of t4. the exponent of U is equal to the order of tMoreover g may be chosen to have prime power order if t is in the center of T, and g may be chosen in T otherwise.A simple corollary is that an element t in T is not in Z*(G) if and only if there is some s ≠ t such that s and tcommute and s and t are G conjugate.A generalization to odd primes was recorded in (Guralnick & Robinson 1993): if t is an element of prime order p andthe commutator [t,g] has order coprime to p for all g, then t is central modulo the p′-core. This was also generalizedto odd primes and to compact Lie groups in (Mislin & Thévenaz 1991), which also contains several useful results inthe finite case.

References• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman,

Graham, Finite simple groups. Proceedings of an Instructional Conference organized by the LondonMathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: AcademicPress, pp. 249–327, ISBN 978-0-12-563850-0, MR0360785 gives a detailed proof of the Brauer–Suzuki theorem.

• Glauberman, George (1966), "Central elements in core-free groups", Journal of Algebra 4: 403–420,doi:10.1016/0021-8693(66)90030-5, ISSN 0021-8693, MR0202822, Zbl 0145.02802

• Guralnick, Robert M.; Robinson, Geoffrey R. (1993), "On extensions of the Baer-Suzuki theorem", Israel Journalof Mathematics 82 (1): 281–297, doi:10.1007/BF02808114, ISSN 0021-2172, MR1239051, Zbl 0794.20029

• Mislin, Guido; Thévenaz, Jacques (1991), "The Z*-theorem for compact Lie groups", Mathematische Annalen291 (1): 103–111, doi:10.1007/BF01445193, ISSN 0025-5831, MR1125010

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Zassenhaus lemma 221

Zassenhaus lemma

Hasse diagram of the Zassenhaus "butterfly" lemma - smaller subgroups aretowards the top of the diagram

In mathematics, the butterfly lemma orZassenhaus lemma, named after HansJulius Zassenhaus, is a technical result onthe lattice of subgroups of a group or thelattice of submodules of a module, or moregenerally for any modular lattice.[1]

Lemma: Suppose is a group withoperators and and are subgroups.Suppose

and are stable subgroups. Then,

is isomorphic to Zassenhaus proved this lemma specifically to give the smoothest proof of the Schreier refinement theorem. The'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved.

Notes[1] See Pierce, p. 27, exercise 1.

References• Pierce, R. S., Associative algebras, Springer, pp. 27, ISBN 0387906932.• Goodearl, K. R.; Warfield, Robert B. (1989), An introduction to noncommutative noetherian rings, Cambridge

University Press, pp. 51, 62, ISBN 9780521369251.• Lang, Serge, Algebra, Graduate Texts in Mathematics (Revised 3rd ed.), Springer-Verlag, pp. 20–21,

ISBN 9780387953854.• Carl Clifton Faith, Nguyen Viet Dung, Barbara Osofsky. Rings, Modules and Representations. p. 6. AMS

Bookstore, 2009. ISBN 0821843702

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Zassenhaus lemma 222

External links• Zassenhaus Lemma and proof at http:/ / www. artofproblemsolving. com/ Wiki/ index. php/

Zassenhaus%27s_Lemma

ZJ theoremIn mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable andhas a normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup of G, for any Sylowp-subgroup S.

Notation and definitions• J(S) is the Thompson subgroup of a p-group S: the subgroup generated by the abelian subgroups of maximal

order.• Z(H) means the center of a group H.• Op′ is the maximal normal subgroup of G of order coprime to p, the p′-core• Op is the maximal normal p-subgroup of G, the p-core.• Op′,p(G) is the maximal normal p-nilpotent subgroup of G, the p′,p-core, part of the upper p-series.• For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-stable if whenever P is a p-subgroup of G such that

POp′(G) is normal in G, and [P,x,x] = 1, then the image of x in NG(P)/CG(P) is contained in a normal p-subgroupof NG(P)/CG(P).

• For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-constrained if the centralizer CG(P) is contained inOp′,p(G) whenever P is a Sylow p-subgroup of Op′,p(G).

References• Glauberman, George (1968), "A characteristic subgroup of a p-stable group" [1], Canadian Journal of

Mathematics 20: 1101–1135, ISSN 0008-414X, MR0230807• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR81b:20002• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra 13:

149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR0245683

References[1] http:/ / www. cms. math. ca/ cjm/ v20/ p1101

Page 228: Theorems in Algebra

Article Sources and Contributors 223

Article Sources and ContributorsAbel's binomial theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612000  Contributors: Brilliant trees, Cronholm144, Geometry guy, Giftlite, Kiensvay, Ktr101, Michael Hardy,PrimeHunter, Xanthoxyl, 5 anonymous edits

Abel–Ruffini theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455715455  Contributors: Albmont, Andrewpmk, AxelBoldt, BeteNoir, BigDom, CRGreathouse, Cf. Hay, CharlesMatthews, Cheesefondue, Colonies Chris, Crasshopper, Dmharvey, Doctormatt, Dominus, Duoduoduo, Edudobay, Ettrig, EverGreg, Fibonacci, Gene Nygaard, Geschichte, Giftlite, Helder.wiki,IRP, Icairns, Inquam, JackSchmidt, JamesBWatson, Kutu su, LamilLerran, Lunae, Marc Venot, Marc van Leeuwen, Mets501, Mh, Michael Hardy, Mike4ty4, Nbarth, Nsh, Oleg Alexandrov,Paddles, Psychonaut, Quantpole, RDBury, Rlinfinity, RodC, Salix alba, Sandrobt, Schneelocke, Simetrical, Stormwyrm, Sue Gardner, Swordsmankirby, Tbjablin, Thegeneralguy, Tide rolls,Ttwo, VKokielov, XJamRastafire, Zchenyu, 51 anonymous edits

Abhyankar's conjecture  Source: http://en.wikipedia.org/w/index.php?oldid=455857476  Contributors: Bender235, CRGreathouse, Charles Matthews, David Haslam, Giftlite, Mackdiesel5,MathMartin, Plclark, R.e.b., Rschwieb, 2 anonymous edits

Acyclic model  Source: http://en.wikipedia.org/w/index.php?oldid=455866493  Contributors: EmanWilm, Giftlite, GregorB, Inarchus, Jitse Niesen, Michael Hardy, Ringspectrum, Sodin,Techman224, Vaughan Pratt, 7 anonymous edits

Ado's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612088  Contributors: CRGreathouse, Charles Matthews, Geometry guy, Giftlite, Jitse Niesen, Marc van Leeuwen, R.e.b.,TakuyaMurata, Tarret, Tesseran, ЈусуФ, 1 anonymous edits

Alperin–Brauer–Gorenstein theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455629144  Contributors: Algebraist, Davcrav, Giftlite, Headbomb, JackSchmidt, KathrynLybarger,Messagetolove, Nbarth, Sodin, Vanish2

Amitsur–Levitzki theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455760783  Contributors: Darij, Giftlite, Headbomb, Michael Hardy, R.e.b., Rschwieb

Artin approximation theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455767217  Contributors: Beetstra, BeteNoir, Charles Matthews, Gauge, Giftlite, Jowa fan, Oleg Alexandrov,Sodin, 9 anonymous edits

Artin–Wedderburn theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612164  Contributors: BeteNoir, Charles Matthews, Darij, Geometry guy, Giftlite, JamieVicary, John Baez,Justpasha, Psychonaut, Qutezuce, Rgdboer, Rschwieb, The Anome, Vivacissamamente, Waltpohl, 20 anonymous edits

Artin–Zorn theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609464  Contributors: David Eppstein, Giftlite, Headbomb, Michael Hardy, Sodin, Tobias Bergemann, Waltpohl, 1anonymous edits

Baer–Suzuki theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455763227  Contributors: Gregbard, Headbomb, Krasnoludek, R.e.b., Rschwieb

Beauville–Laszlo theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455859427  Contributors: Giftlite, Jowa fan, Michael Hardy, R.e.b., Ryan Reich, Sodin, 1 anonymous edits

Binomial inverse theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612331  Contributors: Ballista121, CBM, Cmansley, Entropeneur, Geometry guy, Giftlite, Headbomb,JRSpriggs, Jmath666, Jonas August, MaxSem, Michael Hardy, Neparis, O18, Robinh, TedPavlic, Thecheesykid, 3 anonymous edits

Binomial theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612424  Contributors: -Ozone-, .:Ajvol:., 0, Acdx, Aciel, Adashiel, Aj.sujit, AlanUS, Alansohn, Andres, AnonymousDissident, Antandrus, Arthur Rubin, AxelBoldt, Babababoshka, Badgernet, BeteNoir, Bharatveer, BiT, Binary TSO, Blaxthos, Bombshell, Borisblue, Bsodmike, CRGreathouse, Carbuncle,Cdang, Charibdis, Charles Matthews, Chemninja, Chuchogl, Conversion script, Cpoynton, Crabula, Cronholm144, DEMcAdams, Danno uk, Dirkbb, Discospinster, Dmcq, Duncharris, EbonyJackson, EcneicsFlogCitanaf, Elockid, Eric119, FactSpewer, FlamingSilmaril, Geometry guy, Gesslein, Giants27, Giftlite, Gilliam, Goldencako, Gombang, Gregbard, Gsmgm, Gurch, Hahamhanuka, Hawk8103, ILovePlankton, Iridescent, J.delanoy, JForget, Jagged 85, Jay Gatsby, Jim.belk, Joe056, JorgeGG, Jusdafax, Justin W Smith, KIMWOONGJI, Kbdank71, Keilana, Kenyon,Kiensvay, Kingpin13, Kloddant, Knight1993, La goutte de pluie, Leyo, Linas, Loadmaster, LouScheffer, Lupin, MC10, MER-C, MSGJ, Marc van Leeuwen, MarkSweep, MarnetteD, Matevzk,Meaghan, Meldor, Mets501, Michael Hardy, Michael Slone, Mikez, Minthellen, Molotron, Myrizio, NOrbeck, Najoj, Nbarth, Nirvana89, Nk, Nomet, Nonagonal Spider, Oceanwitch, OlegAlexandrov, Opabinia regalis, PMDrive1061, PTP2009, PericlesofAthens, Pevernagie, Pion, Pleasantville, Poetaris, Poor Yorick, Quietbritishjim, R.e.b., RMFan1, RexNL, Rich Farmbrough,RickDC, Robinh, Salvatore Ingala, Saric, Sbealing, Shahab, SimonP, StandardizerII, Stebulus, Sławomir Biały, THEN WHO WAS PHONE?, TNeloms, TakuyaMurata, Tbsmith, Tcnuk, TheThing That Should Not Be, The absolute real deal, TheSuave, Thetruthseer, Tobias Bergemann, Ulfalizer, Urdutext, Verdy p, WardenWalk, Warut, Wiki alf, Wings Upon My Feet, Wroscel,Zchenyu, Zvika, 325 anonymous edits

Birch's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455796486  Contributors: Akrabbim, Charles Matthews, Mon4, Sodin, 1 anonymous edits

Birkhoff's representation theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612525  Contributors: Chris the speller, David Eppstein, Gabelaia, Geometry guy, Giftlite,Headbomb, Igorpak, Mhym

Boolean prime ideal theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612600  Contributors: Aleph4, AugPi, AxelBoldt, CBM, CRGreathouse, Charles Matthews, Chinju, DavidEppstein, Dysprosia, Eric119, Geometry guy, Giftlite, Headbomb, Hennobrandsma, Hugo Herbelin, Jon Awbrey, Kope, MarkSweep, Markus Krötzsch, Mhss, Michael Hardy, Ott2, RobHar,RoodyAlien, TexD, Tkuvho, Tobias Bergemann, Trioculite, Trovatore, Vivacissamamente, 15 anonymous edits

Borel–Weil theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612643  Contributors: Algebraist, Arcfrk, David Eppstein, Geometry guy, Giftlite, Michael Hardy, Nicolaisvendsen,Psychonaut, SemperBlotto, Tabletop, 1 anonymous edits

Borel–Weil–Bott theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455612716  Contributors: Algebraist, BeteNoir, Bunny Angel13, Charles Matthews, Fropuff, Gauge, GeneNygaard, Geometry guy, Giftlite, M m hawk, Masnevets, Michael Hardy, Myasuda, Nbarth, Psychonaut, RobHar, Silly rabbit, Waltpohl, 4 anonymous edits

Brauer's theorem on induced characters  Source: http://en.wikipedia.org/w/index.php?oldid=455614459  Contributors: BeteNoir, Charles Matthews, Geffrey, Geometry guy, Giftlite,JackSchmidt, Masnevets, Messagetolove, 5 anonymous edits

Brauer's three main theorems  Source: http://en.wikipedia.org/w/index.php?oldid=455614539  Contributors: Arcfrk, Charles Matthews, Davcrav, DavidCBryant, Geometry guy, Headbomb,Messagetolove, Oleg Alexandrov, R.e.b., Thecheesykid

Brauer–Cartan–Hua theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455614613  Contributors: 1ForTheMoney, Geometry guy, Headbomb, JackSchmidt, Omnipaedista,PMDrive1061, Pillsberry, Pmanderson, Sławomir Biały, THEN WHO WAS PHONE?

Brauer–Nesbitt theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455614718  Contributors: Charles Matthews, Geometry guy, Michael Slone, R.e.b., 2 anonymous edits

Brauer–Siegel theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609585  Contributors: 3mta3, Charles Matthews, Giftlite, JackSchmidt, Michael Hardy, Sodin, Vanish2, Vesath

Brauer–Suzuki theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609550  Contributors: Bird of paradox, Charles Matthews, Davcrav, Headbomb, JackSchmidt, Kmhkmh, R.e.b.,Sodin, 1 anonymous edits

Brauer–Suzuki–Wall theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609607  Contributors: Gregbard, Headbomb, Michael Hardy, R.e.b., Slawekb, Sodin

Burnside theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455614918  Contributors: Ashsong, BeteNoir, Charles Matthews, Gebstadter, Geometry guy, Giftlite, Lucifer Anh,Messagetolove, Oleg Alexandrov, SchfiftyThree, Tobias Bergemann, 12 anonymous edits

Cartan's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455619115  Contributors: Asmeurer, BeteNoir, Charles Matthews, Fropuff, Geometry guy, Giftlite, Headbomb, JCSantos,R'n'B, Silly rabbit, Sławomir Biały, Topology Expert, Unyoyega, Zoran.skoda

Cartan–Dieudonné theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455615149  Contributors: Charles Matthews, Crazy runner, Geometry guy, Keyi, MathMartin, Michael Hardy,Nbarth, Oleg Alexandrov, Silly rabbit, 2 anonymous edits

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Cauchy's theorem (group theory)  Source: http://en.wikipedia.org/w/index.php?oldid=455615270  Contributors: AdamSmithee, AlexDenney, Algebraist, Andrei G Kustov, Blotwell,Egpetersen, Eric119, Fell Collar, Geometry guy, Giftlite, JackSchmidt, Kilva, LOL, Michael Hardy, Nm420, PV=nRT, R'n'B, Rghthndsd, Scottie 000, TakuyaMurata, Tango, Vanish2, 11anonymous edits

Cayley's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455615349  Contributors: Ambarish, Atyatya, AxelBoldt, BeteNoir, Calle, Charles Matthews, Chas zzz brown,ChickenMerengo, David Pierce, Dbenbenn, Dominus, Dr Zimbu, Dysprosia, Geometry guy, Giftlite, Hairy Dude, Headbomb, Helder.wiki, Inquam, JackSchmidt, Jane Bennet, Joen235,Joshuagmath, Looxix, Maxal, Mhss, Michael Hardy, Myasuda, Patrick, Scubbo, Simetrical, Thehotelambush, Tosha, Wapcaplet, Zvika, 38 anonymous edits

Cayley–Hamilton theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455615451  Contributors: Albmont, Alexander Chervov, Algebraist, Aliotra, Attilios, AxelBoldt, BeteNoir,CRGreathouse, Carbuncle, Charles Matthews, Chetvorno, Choco.litt, Closedmouth, Cuzkatzimhut, David Shay, David Sneek, Dominus, Dysprosia, Eecs student, Equendil, Gene Nygaard,Geometry guy, Giftlite, Headbomb, Ivan Štambuk, Jcobb, JdH, Jpkotta, Jujutacular, Lambiam, Lechatjaune, LilHelpa, Ling.Nut, Lionelbrits, Looxix, Malatinszky, Marc van Leeuwen,MathKnight, MehdiPedia, Merrybrit, Michael Hardy, Nschoem, NyAp, Obradovic Goran, Oleg Alexandrov, PAR, PhilDWraight, Pmdboi, Psychonaut, Quasicharacter, Rik Bos, Ryan Reich,SDaniel, Shreevatsa, Sławomir Biały, Tarquin, The suffocated, Turgidson, Txomin, Werdenwissen, Wshun, 82 anonymous edits

Chevalley–Shephard–Todd theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455616003  Contributors: Arcfrk, Closedmouth, Geometry guy, Giftlite, Headbomb, Michael Hardy,R.e.b., Rjwilmsi, 4 anonymous edits

Chevalley–Warning theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455767620  Contributors: BeteNoir, Charles Matthews, David Brink, David Eppstein, Dtrebbien, Giftlite,Headbomb, JackSchmidt, Jowa fan, LilHelpa, Michael Hardy, R.e.b., Sodin, 5 anonymous edits

Classification of finite simple groups  Source: http://en.wikipedia.org/w/index.php?oldid=455616173  Contributors: 63.162.153.xxx, 66.38.184.xxx, A3 nm, Aboriginal Noise, Akriasas, Andi5,Arneth, Astronautics, AxelBoldt, BananaFiend, Bruguiea, C S, Calabraxthis, Cambyses, Charles Matthews, Chas zzz brown, Conversion script, Dominus, Dratman, Drschawrz, Dysprosia,Eugene van der Pijll, Gareth Jones, Gene Ward Smith, Geometry guy, Giftlite, Gilliam, Gro-Tsen, Huppybanny, JackSchmidt, Jim.belk, Jitse Niesen, John of Reading, JoshuaZ, Jowa fan, Joy,Kidburla, Kilva, Lightmouse, Looxix, Loren Rosen, Maproom, Matt Crypto, Melchoir, Messagetolove, Michael Hardy, Michael Larsen, Mike40033, Nbarth, PierreAbbat, Pjacobi, Poulpy, R.e.b.,Rjwilmsi, Salix alba, Schneelocke, Slightsmile, Tarquin, Timwi, Tobias Bergemann, Topbanana, Twri, VeryVerily, VladimirReshetnikov, Zundark, 59 anonymous edits

Cohn's irreducibility criterion  Source: http://en.wikipedia.org/w/index.php?oldid=455860476  Contributors: Cornflake pirate, David Eppstein, Dtrebbien, EdJohnston, Gandalf61, Giftlite,JaGa, John Vandenberg, Ntsimp, Oleg Alexandrov, Omid Hatami, RHaworth, Sodin, Ultimus, 7 anonymous edits

Cramer's rule  Source: http://en.wikipedia.org/w/index.php?oldid=455616575  Contributors: A19grey, AlanUS, Alansohn, Alphax, Anonymous Dissident, Arthur Rubin, Asmeurer, Barak Sh,Ben Spinozoan, Bender2k14, Benzi455, BeteNoir, Bh3u4m, Carmichael95, Ciphers, Cloudguitar, Cryptic, D0762, Dingenis, Djwinters, Dmaher, ERcheck, EconoPhysicist, Elphion, Epbr123,Flavio Guitian, FlowRate, Foxjwill, Franklin.vp, Gauge, Geometry guy, GermanX, Giftlite, Grafen, Hillbrand, Ioscius, J1e9n8s5, JHMM13, JakeVortex, Javanbakht, Jaysweet, Jcobb, Jks,Jmath666, Joriki, KathrynLybarger, Khabgood, Koavf, LOL, Lear's Fool, LilHelpa, Lleeoo, Lradrama, Malo, Marc van Leeuwen, Maxno, Mebden, Michael Hardy, Michael Slone, Mike Rosoft,Mormegil, Nemolus77, Obradovic Goran, Oleg Alexandrov, Oli Filth, Ooz dot ie, Patrick, Pharaoh of the Wizards, Protonk, Rludlow, Rogper, Saravask, Serebr, Silly rabbit, Snailwalker, Spoon!,Steve.jaramillov, Sławomir Biały, Tarquin, TedPavlic, TheMaestro, Thumperward, TotientDragooned, Tristanreid, User A1, Vivacissamamente, Vsb, Waggers, Wdspann, Wolfrock, Xaos,Yecril, Yoshiki Sunada, Yossiea, Zipcodeman, 163 anonymous edits

Crystallographic restriction theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455616670  Contributors: Armando-Martin, Ben Standeven, BeteNoir, Bgm2011, Charles Matthews,Commander Keane, Dyaa, Euyyn, Geometry guy, Giftlite, Greg Kuperberg, Headbomb, Helder.wiki, Howard Landman, Joseph Myers, Jowa fan, KSmrq, Keenan Pepper, Michael Hardy,Mvpranav, NRLer, Oleg Alexandrov, Paolo.dL, Patrick, Rjwilmsi, Tesscass, Unco, Zvika, 16 anonymous edits

Descartes' rule of signs  Source: http://en.wikipedia.org/w/index.php?oldid=455616871  Contributors: 478jjjz, A. Pichler, Adam Field, Alansohn, Alethiareg, Bender235, Bento00, CharlesMatthews, Chenxlee, Chuunen Baka, DavidMcKenzie, Dzordzm, Estudiarme, FHGJ, GNB, Gandalf61, Gazpacho, Geometry guy, Giftlite, Haihe, Jeekc, Jeepday, JimVC3, Lechatjaune, LouCrazy, Magic Window, Mets501, Mglg, Michael Hardy, Mild Bill Hiccup, Nishantsah, PV=nRT, Palladinus, Pdebart, Reywas92, Salix alba, Silly rabbit, Sinblox, Spacepotato, Tide rolls, Tosha,Vroo, 51 anonymous edits

Dirichlet's unit theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609753  Contributors: 4meter4, Bender235, BeteNoir, CRGreathouse, Charles Matthews, CharlesGillingham,Chenxlee, Crisófilax, Fropuff, Gene Ward Smith, Giftlite, Headbomb, Michael Hardy, R.e.b., Revolver, Rich Farmbrough, Ringspectrum, RobHar, Sodin, Vanish2, Vargenau, 11 anonymous edits

Engel theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455801855  Contributors: Arcfrk, BeteNoir, CSTAR, Charles Matthews, Darij, Geometry guy, Giftlite, Headbomb,JackSchmidt, Mets501, Michael Hardy, Nbarth, R.e.b., Safemariner, Wjcook, 9 anonymous edits

Factor theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455694654  Contributors: Arthena, Balrog30, Barak, BeteNoir, Bsadowski1, Celestianpower, Charles Matthews, ChuunenBaka, David Radcliffe, Discospinster, Geometry guy, Giftlite, Iain.dalton, Jacj, Kiensvay, Kurosuke88, LOL, La Pianista, La goutte de pluie, Magister Mathematicae, Mairi, MarSch, Mpatel,Qwfp, RMFan1, S243a, Static shock1994, Tooto, 59 anonymous edits

Feit–Thompson theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609773  Contributors: AndrewWTaylor, BeteNoir, Bird of paradox, Charles Matthews, Gauge, Giftlite,JackSchmidt, Jim.belk, Kilva, Linas, Malatinszky, Messagetolove, Michael Hardy, Nicholas Jackson, PerryTachett, Psychonaut, R.e.b., Rjwilmsi, Sodin, Sullivan.t.j, Tyomitch, Woohookitty,Zundark, 9 anonymous edits

Fitting's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455624594  Contributors: CBM, Charles Matthews, Geometry guy, MarSch, Michael Hardy, Nbarth, Paul Klenk, R.e.b.,Zundark

Focal subgroup theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455619355  Contributors: Giftlite, Headbomb, JHunterJ, JackSchmidt, Michael Hardy, Nbarth, Rschwieb

Frobenius determinant theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455619264  Contributors: Bearcat, David Eppstein, Dd314, Ebe123, Geometry guy, Giftlite, Headbomb,LilHelpa, Michael Hardy, Pichpich, R.e.b., Sadads

Frobenius theorem (real division algebras)  Source: http://en.wikipedia.org/w/index.php?oldid=455618771  Contributors: BartekChom, Charles Matthews, Geometry guy, Giftlite, Incnis Mrsi,JAMnx, Jim.belk, JukoFF, KSmrq, MarkSweep, MarkusQ, MathMartin, Michael Hardy, Mild Bill Hiccup, OneWeirdDude, Rgdboer, RobHar, WestwoodMatt, 17 anonymous edits

Fundamental lemma (Langlands program)  Source: http://en.wikipedia.org/w/index.php?oldid=455796915  Contributors: Arcfrk, Charles Matthews, David Eppstein, Dominus, Giftlite,Headbomb, LJosil, Michael Hardy, Phanjuy, RobHar, Rumping, Slawekb, Sodin, Stevey7788, Sławomir Biały, TakuyaMurata, Tobias Bergemann, 11 anonymous edits

Fundamental theorem of algebra  Source: http://en.wikipedia.org/w/index.php?oldid=455618556  Contributors: .:Ajvol:., 64.12.102.xxx, Abovechief, Adam majewski, Ahoerstemeier,Alansohn, [email protected], Algebraist, Alink, Andy Fugard, Archelon, Arthena, Arthur Rubin, Arved, Aude, AugPi, AxelBoldt, BeteNoir, Bidabadi, BigJohnHenry, Blindsuperhero,Bob.v.R, Can't sleep, clown will eat me, Charles Matthews, Charleswallingford, Conversion script, Cybercobra, Daran, Darij, Deineka, Dmn, DonSiano, Doradus, Drilnoth, Dysprosia, EmilJ, Evilsaltine, Fredrik, Furrykef, Gaius Cornelius, Gene Ward Smith, Geometry guy, Giftlite, Graham87, Greg Kuperberg, Hede2000, Helix84, Henning Makholm, Hesam7, Holger Blasum,Huddlebum, Icairns, JCSantos, Jacobolus, JdH, Jimbreed, Jimbryho, Jimp, Jóna Þórunn, Kartik J, Lakinekaki, Lambiam, Li-sung, LkNsngth, Lunchscale, Lupin, LutzL, MathMartin, MathsIsFun,Mav, Meni Rosenfeld, Michael Hardy, Michael Larsen, Michael Slone, Mike Segal, Monamip, Mpatel, Nic bor, Nsh, Nuno Tavares, Obradovic Goran, Ortonmc, Oxy86, PMajer, Paul D.Anderson, Paul Taylor, Philologer, Primalbeing, Pt, Qmwne235, Randomblue, Rgdboer, Rholton, Rich Farmbrough, Rjwilmsi, Robinh, Romanm, Shishir0610, Skomorokh, Smcinerney,Smimram, Snoyes, SoroSuub1, Syp, Tobias Bergemann, Toby Bartels, Toh, Trovatore, Tulcod, Unyoyega, Vladkornea, WikiUserPedia, Wmahan, Woohookitty, Wshun, XJamRastafire,Xantharius, Zfr, Zundark, Zvika, 109 anonymous edits

Fundamental theorem of cyclic groups  Source: http://en.wikipedia.org/w/index.php?oldid=455618448  Contributors: Arthur Rubin, Cybercobra, Geometry guy, Giftlite, Icairns, Joeldl, Lhf,Michael Hardy, Selfworm, Sigmundur, Tobias Bergemann, Zvika, 20 anonymous edits

Fundamental theorem of Galois theory  Source: http://en.wikipedia.org/w/index.php?oldid=455618329  Contributors: Bender235, Bender2k14, BeteNoir, Charles Matthews, Cwkmail,Cybercobra, Dfeuer, Dmharvey, Dysprosia, Frau Holle, Geometry guy, Giftlite, Hesam7, HorsePunchKid, Icairns, Jibbb, Jim.belk, MarkC77, MathMartin, Sandrobt, 15 anonymous edits

Fundamental theorem of linear algebra  Source: http://en.wikipedia.org/w/index.php?oldid=455618237  Contributors: BeteNoir, Charles Matthews, Cronholm144, Cybercobra, Flavio Guitian,Geometry guy, Giftlite, Harryboyles, Icairns, Keenan Pepper, Lowellian, Nbarth, Oleg Alexandrov, Qwfp, R'n'B, Silly rabbit, 17 anonymous edits

Fundamental theorem on homomorphisms  Source: http://en.wikipedia.org/w/index.php?oldid=455618150  Contributors: Andre Engels, Arthena, AugPi, AxelBoldt, BeteNoir, Charles Matthews, Chas zzz brown, Conversion script, Cybercobra, DefLog, Erud, Gelingvistoj, Geometry guy, Giftlite, Goochelaar, Graham87, Grubber, Icairns, Jay Gatsby, Linas, Magidin,

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MathMartin, Waltpohl, Weialawaga, 9 anonymous edits

Gilman–Griess theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455766095  Contributors: Giftlite, Headbomb, Michael Hardy, R.e.b., Rjwilmsi, Sodin, 1 anonymous edits

Going up and going down  Source: http://en.wikipedia.org/w/index.php?oldid=455619435  Contributors: Charles Matthews, David Shay, Discospinster, Giftlite, Jakob.scholbach, Jowa fan,Kiefer.Wolfowitz, Michael Slone, Paul August, Revolver, Rschwieb, 11 anonymous edits

Goldie's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455618039  Contributors: Charles Matthews, Geometry guy, Giftlite, JackSchmidt, Jowa fan, Rgdboer, Rschwieb, TobiasBergemann, Vanish2

Golod–Shafarevich theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455617950  Contributors: Academic Challenger, Bender235, Charles Matthews, Ebony Jackson, Geometryguy, Giftlite, JackSchmidt, Jackbarron, Mathsci, Michael Hardy, Michael Slone, RobHar, Turgidson, Yonatbe5, 1 anonymous edits

Gorenstein–Harada theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455617855  Contributors: Geometry guy, R.e.b.

Gromov's theorem on groups of polynomial growth  Source: http://en.wikipedia.org/w/index.php?oldid=455619473  Contributors: BeteNoir, CSTAR, Charles Matthews, Efjb2, Geometry guy,Giftlite, Headbomb, Icairns, Jevansen, JoshuaZ, LarRan, Michael Hardy, Mosher, OdedSchramm, Teorth, The Anome, Tkuvho, Tosha, Zundark, 4 anonymous edits

Grushko theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455766207  Contributors: Colonies Chris, Daniel5Ko, David Eppstein, Giftlite, HUnTeR4subs, JackSchmidt, Katzmik,Michael Hardy, Nsk92, RonnieBrown, Sodin, 2 anonymous edits

Haboush's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765461  Contributors: BeteNoir, Charles Matthews, David Eppstein, DavidCBryant, Giftlite, Jeff3000, R'n'B, R.e.b.,Ringspectrum, Rjwilmsi, RobHar, Sodin, 4 anonymous edits

Hahn embedding theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455619581  Contributors: BeteNoir, Charles Matthews, Diskz, Gene Ward Smith, Geometry guy, Giftlite,Iohannes Animosus, JackSchmidt, Marcus Pivato, 1 anonymous edits

Hajós's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765482  Contributors: Cronholm144, Giftlite, Kope, PigFlu Oink, R.e.b., Sodin, Tholly, Zachanter

Harish-Chandra isomorphism  Source: http://en.wikipedia.org/w/index.php?oldid=455629394  Contributors: Arcfrk, Charles Matthews, Franp9am, Giftlite, Headbomb, R.e.b., Sodin, Ulner, 6anonymous edits

Hasse norm theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455629296  Contributors: Arcfrk, BeteNoir, Charles Matthews, Dugwiki, Gene Ward Smith, Sodin, Vanish2, 1anonymous edits

Hasse–Arf theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765535  Contributors: Chenxlee, Giftlite, RobHar, Sodin, TakuyaMurata, Woohookitty, 2 anonymous edits

Hilbert's basis theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609894  Contributors: Ad4m, Arcfrk, AxelBoldt, BeteNoir, Brockert, Bryan Derksen, Charles Matthews,Conversion script, Drusus 0, Gaius Cornelius, Giftlite, Guardian of Light, Hillman, ICPalm, Jxr, LilHelpa, MathMartin, Michael Slone, Oleg Alexandrov, R.e.b., Randomblue, Sodin, TobiasBergemann, Vivacissamamente, Waltpohl, 17 anonymous edits

Hilbert's irreducibility theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609923  Contributors: Aleph4, BUF4Life, Barylior, BeteNoir, Brookie, Charles Matthews, Giftlite,Mathtyke, Michael Hardy, Pearle, PoolGuy, Ringspectrum, Sodin, 3 anonymous edits

Hilbert's Nullstellensatz  Source: http://en.wikipedia.org/w/index.php?oldid=455690802  Contributors: Arcfrk, Atgnclk, AxelBoldt, BeteNoir, Bomazi, Charles Matthews, Chas zzz brown,Crisófilax, D.Lazard, D6, EmilJ, Fluxions, Fropuff, Giftlite, Jmath666, Michael Hardy, Mlm42, Nbarth, Ntsimp, Oleg Alexandrov, R.e.b., Ringspectrum, Rschwieb, Sannse, TomyDuby,Trevorgoodchild, Vanish2, Waltpohl, 21 anonymous edits

Hilbert's syzygy theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609953  Contributors: Arcfrk, BeteNoir, Charles Matthews, Giftlite, Grendelkhan, Myrizio, Nick Number,Sodin, Ylloh, Zaphod Beeblebrox, 12 anonymous edits

Hilbert's Theorem 90  Source: http://en.wikipedia.org/w/index.php?oldid=455610006  Contributors: Bender235, BeteNoir, CRGreathouse, Charles Matthews, EmilJ, Four Dog Night, GaiusCornelius, Gene Ward Smith, Giftlite, MathMartin, Myasuda, Pmanderson, R.e.b., Rich Farmbrough, Ringspectrum, RobHar, Set theorist, Sodin, Zundark, 13 anonymous edits

Hopkins–Levitzki theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455618651  Contributors: Anne Bauval, Bearcat, Giftlite, Michael Hardy, R.e.b., Rschwieb

Hurwitz's theorem (normed division algebras)  Source: http://en.wikipedia.org/w/index.php?oldid=455687589  Contributors: 3mta3, Algebraist, Anne Bauval, Arrataz, Brews ohare, Geometryguy, Giftlite, Headbomb, Ilmari Karonen, JCSantos, Mattbuck, RobHar, 15 anonymous edits

Isomorphism extension theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455687893  Contributors: AtticusRyan, Cronholm144, Geometry guy, Giftlite, ImPerfectHacker, MichaelHardy, Vanish2, Zundark, 3 anonymous edits

Isomorphism theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455688042  Contributors: AdamSmithee, Algebraist, Amdk8800, Avani, AxelBoldt, BeteNoir, Brighterorange,Bruguiea, Bryan Derksen, Csigabi, Cwkmail, Dysprosia, EmilyPeters, Flamingspinach, Frodo, Frozsyn, Geometry guy, Giftlite, Grubber, Helder.wiki, Isnow, JackSchmidt, Jcobb, Jeepday,JensMueller, Jim.belk, Karl-Henner, Konradek, Krasnoludek, Lethe, Magidin, MathMartin, Michael Hardy, Michael K. Edwards, Miyagawa, Mrajpkc, Nbarth, Rjgodoy, SJP, Sabbut, SetaLyas,Shiyang, Silly rabbit, Tarquin, Toby, Waltpohl, Weialawaga, Zundark, 41 anonymous edits

Jacobson density theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455697398  Contributors: Bender235, Charles Matthews, David Eppstein, Giftlite, JamieVicary, Mct mht, MikePeel, Philosopher, Point-set topologist, R.e.b., Rjwilmsi, Rschwieb, Tobias Bergemann, Vanish2, Waltpohl

Jordan's theorem (symmetric group)  Source: http://en.wikipedia.org/w/index.php?oldid=455688643  Contributors: Franp9am, Geometry guy

Jordan–Schur theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455688735  Contributors: David Eppstein, Geometry guy, Giftlite, JoshuaZ, Matt me, Michael Hardy, 4 anonymousedits

Krull's principal ideal theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455690389  Contributors: Anne Bauval, BeteNoir, Charles Matthews, Geometry guy, Giftlite,Jakob.scholbach, Kummini, Michael Hardy, Oleg Alexandrov, Silverfish, Vanish2, Vivacissamamente, 3 anonymous edits

Krull–Schmidt theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455695593  Contributors: Aholtman, Algebraist, Arcfrk, Arthur Rubin, Bogdangiusca, Charles Matthews,Dreadstar, Foobarnix, Fropuff, Gauge, Giftlite, Helder.wiki, JackSchmidt, Kevin Lamoreau, Masnevets, Matthew Fennell, Michael Hardy, Oleg Alexandrov, Omnipaedista, Rschwieb, TheAnome, Tobias Bergemann, Zundark, 19 anonymous edits

Künneth theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455744254  Contributors: Ambrose H. Field, Auntof6, Bender235, Charles Matthews, Cheesus, DVD R W, Dbenbenn,Giftlite, Gofors, KSmrq, LokiClock, Marhahs, Momotaro, Old Man Grumpus, Ozob, R.e.b., Rschwieb, Ryan Reich, Silly rabbit, 16 anonymous edits

Kurosh subgroup theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455630389  Contributors: Dtrebbien, Giftlite, HUnTeR4subs, JackSchmidt, LilHelpa, Michael Hardy, Nsk92,Sodin, 3 anonymous edits

Lagrange's theorem (group theory)  Source: http://en.wikipedia.org/w/index.php?oldid=455610063  Contributors: Aboalbiss, AdamSmithee, Avik21, AxelBoldt, BeteNoir, Bryan Derksen,Calle, Charles Matthews, Chas zzz brown, Chromosome, Conversion script, Courcelles, Creidieki, Cwkmail, Dcoetzee, Dysprosia, Eric119, Giftlite, Goochelaar, Graham87, GregorB, Grubber,Hyju, Ixfd64, JCSantos, JackSchmidt, Joth, Kilva, Leycec, Lhf, Lowellian, Lupin, Mathsci, Miaow Miaow, Michael Hardy, Obradovic Goran, Plasticup, Quotient group, Reaper Eternal,Rghthndsd, Salvatore Ingala, Sodin, Superninja, Tarquin, Timwi, Tsemii, Xantharius, Youandme, Yuval Madar, 40 anonymous edits

Lasker–Noether theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455696265  Contributors: Arthur Rubin, Bender2k14, BeteNoir, Charles Matthews, David Eppstein, Expz, GeneNygaard, Giftlite, Gwaihir, Ioannes Pragensis, Jakob.scholbach, KWRegan, Kummini, MathMartin, Michael Hardy, Mild Bill Hiccup, Nbarth, Oleg Alexandrov, Paisa, Psychonaut, R.e.b., RichFarmbrough, Rschwieb, Silverfish, TakuyaMurata, WATARU, Waltpohl, XPEHOPE3, 14 anonymous edits

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Latimer-MacDuffee theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455884452  Contributors: Charles Matthews, Konstable, Oleg Alexandrov, Sodin, 2 anonymous edits

Lattice theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455717612  Contributors: Algebraist, BeteNoir, Bruguiea, Cacadril, Caesura, Cwkmail, David Eppstein, DemonThing,E946, Error792, Frédérick Lacasse, Giftlite, GregorB, Hans Adler, J•A•K, MathMartin, MathMast, Nbarth, Patrick, RDBury, Silly rabbit, Thehotelambush, Tobias Bergemann, Vanish2, 4anonymous edits

Levitzky's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455630202  Contributors: Calaka, Giftlite, Headbomb, JackSchmidt, Kope, Michael Hardy, Point-set topologist, RichFarmbrough, Rjwilmsi, Sodin, Zundark

Lie's third theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455690122  Contributors: Charles Matthews, FrozenPurpleCube, Geometry guy, Giftlite, Jason Quinn

Lie–Kolchin theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455717004  Contributors: BeteNoir, Ceyockey, Charles Matthews, CrackerJack7891, David Eppstein, FactSpewer,Gauge, Gene Nygaard, Giftlite, Headbomb, Hillman, JackSchmidt, Linas, Michael Hardy, Natalya, Nbarth, Nowhither, Psychonaut, R.e.b., RDBury, West Brom 4ever, Zundark, 4 anonymousedits

Maschke's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455618805  Contributors: Arcfrk, Aviados, BeteNoir, Calaka, CapitalR, Charles Matthews, Cwkmail, Dcoetzee, Fropuff,Giftlite, Gwaihir, Headbomb, Hesam7, Hillman, Jay Gatsby, Jtwdog, Michael Hardy, Natalya, PappyK, RobHar, Rschwieb, Solar-Poseidon, Tobias Bergemann, TommasoT, Undercrowdtroll,Xiaodai, Zundark, 18 anonymous edits

Milnor conjecture  Source: http://en.wikipedia.org/w/index.php?oldid=455716070  Contributors: C S, Charles Matthews, Exceptg, Gauge, Headbomb, Michael Hardy, Phantomsteve, R.e.b.,RDBury, Semorrison, Singularity, 5 anonymous edits

Mordell–Weil theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455630342  Contributors: BeteNoir, Charles Matthews, Giftlite, Hesam7, Jcobb, Patrick, Psychonaut, R.e.b., Reedy,Silenteuphony, Sodin, Thecheesykid, Vanish2, Zoicon5, Zundark, 3 anonymous edits

Multinomial theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455690672  Contributors: AdjustShift, Altenmann, ArglebargleIV, Charles Matthews, Chenxlee, Chessimprov,Dysprosia, Endlessoblivion, Forge021, Geometry guy, Giftlite, Icairns, Jengelh, Kakila, LOL, Labus, Linas, McKay, MikeRumex, NeoUrfahraner, Pberndt, Quantling, Rar, Rich Farmbrough,RokerHRO, SPUI, Spoon!, Stephenb, WiiStation360, Wile E. Heresiarch, Yaleeconometrics, Zero0000, 70 ,נו, טוב anonymous edits

Nielsen–Schreier theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455630131  Contributors: David Eppstein, Giftlite, HUnTeR4subs, Headbomb, JackSchmidt, Sodin, Zundark, 1anonymous edits

Perron–Frobenius theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455759198  Contributors: Alexander Chervov, Arcfrk, B.wilson, BenFrantzDale, Bender235, BeteNoir, Billluv2fly, Charles Matthews, Comfortably Paranoid, Cvdwoest, David Eppstein, Dcclark, Dima373, Doctorilluminatus, Flyingspuds, G.perarnau, Gdm, Giftlite, Justin Mauger, Kiefer.Wolfowitz,Kirbin, Linas, MRFS, Michael Hardy, Nbarth, Pavel Stanley, Psychonaut, R.e.b., Rschwieb, Shining Celebi, Sodin, Stigin, Tcnuk, TedPavlic, Urhixidur, Vinsz, 59 anonymous edits

Poincaré–Birkhoff–Witt theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455690945  Contributors: Arcfrk, AxelBoldt, BeteNoir, Burivykh, CSTAR, DR2006kl, Dan Gardner,Darij, Geometry guy, Giftlite, Hans Lundmark, Henning Makholm, Michael Hardy, Myasuda, Oleg Alexandrov, Psychonaut, R.e.b., Vanished user, 13 anonymous edits

Polynomial remainder theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455691069  Contributors: Aakaalaar93, BeteNoir, Charles Matthews, Corwin., Fangz, Geometry guy,Gesslein, Giftlite, Jusdafax, KSmrq, Kenny TM~, Lambiam, MSGJ, Maxal, Michael Hardy, Mifter, Nonagonal Spider, Oli Filth, Pizza1512, Rommels, Samw, Silverfish, XMxWx, YUL89YYZ,40 anonymous edits

Primitive element theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455691200  Contributors: Algebraist, Austinmohr, Billymac00, Bo Jacoby, Charles Matthews, Dfeldmann, GeneWard Smith, Geometry guy, Giftlite, IhorLviv, MathMartin, Michael Hardy, Oyd11, Point-set topologist, Sandrobt, Simetrical, Vanish2, William Avery, Zundark, 9 anonymous edits

Quillen–Suslin theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455696014  Contributors: BeteNoir, Charles Matthews, Dan Gardner, Dtrebbien, Giftlite, Itai, Jcobb, Joeldl,Michael Hardy, Myasuda, Pmanderson, Psychonaut, RobHar, Rschwieb, Ryan Reich, Silverfish, Singularity, 18 anonymous edits

Rational root theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455691538  Contributors: 0, 478jjjz, 64.24.17.xxx, A. Pichler, ABF, Aisaac, Aleph4, [email protected],Aliotra, Anonymous Dissident, Apeiron, Asmeurer, AxelBoldt, BeteNoir, Blitz9, Charles Matthews, Count Iblis, Culix, Discospinster, Gak, Geometry guy, Giftlite, Henrygb, Jujutacular,Kmhkmh, Lambiam, Lindmere, Marc Venot, Marc van Leeuwen, Mets501, Michael Hardy, N5iln, Nousernamesleft, Postglock, Reyk, RoseParks, Salgueiro, Sam Hocevar, Sikory, Taw, TheCunctator, Tobias Hoevekamp, Zooplankton1972, Zundark, 42 anonymous edits

Regev's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455610675  Contributors: Giftlite, Gregbard, Headbomb, Michael Hardy, R.e.b., Sodin

Schreier refinement theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455691924  Contributors: BeteNoir, CBM, Charles Matthews, David Eppstein, Gauge, Geometry guy,ImPerfectHacker, Jaakko Seppälä, Michael Slone, Spartanfox86, Tobias Bergemann, 1 anonymous edits

Schur–Zassenhaus theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455697217  Contributors: Algebraist, Joe Decker, Kidburla, Marvoir, Mm06ahlf, Rich Farmbrough, Rschwieb,Wafulz, 2 anonymous edits

Serre–Swan theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455697339  Contributors: AxelBoldt, Bender235, Charles Matthews, Gene Nygaard, Giftlite, Hillman, JackSchmidt,JoergenB, John Baez, Matterink, Mct mht, Phys, Point-set topologist, Rausch, Rschwieb, Silly rabbit, Tesseran, Tosha, Waltpohl, 6 anonymous edits

Skolem–Noether theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455695953  Contributors: BeteNoir, Charles Matthews, Gaius Cornelius, Giftlite, JackSchmidt, MathMartin,Psychonaut, Rschwieb, The Rambling Man, Thehotelambush, 10 anonymous edits

Specht's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455865798  Contributors: Bender235, Charles Matthews, Giftlite, Jitse Niesen, Michael Hardy, Sodin, 1 anonymous edits

Stone's representation theorem for Boolean algebras  Source: http://en.wikipedia.org/w/index.php?oldid=455744437  Contributors: Aleph0, Beroal, BeteNoir, Blotwell, CBM, Chalst, Chinju,David Eppstein, Falcor84, Fropuff, Giftlite, JanCK, Kuratowski's Ghost, Linas, Markus Krötzsch, Mhss, Michael Hardy, Naddy, Pjacobi, Porton, R'n'B, R.e.b., Rschwieb, Sharpcomputing,Smack, StevenJohnston, Tkuvho, Trovatore, Tsirel, Vivacissamamente, Zundark, 15 anonymous edits

Structure theorem for finitely generated modules over a principal ideal domain  Source: http://en.wikipedia.org/w/index.php?oldid=455693186  Contributors: Alecobbe, Algebraist,Altrevolte, Anne Bauval, ArnoldReinhold, Arthur Rubin, Auntof6, Bomazi, EmilJ, Expz, Foobarnix, Frap, Geometry guy, Giftlite, Grafen, Henning Makholm, JackSchmidt, Kundor, Marc vanLeeuwen, Michael Hardy, Mulanhua, Nbarth, Oleg Alexandrov, Ozob, ReyBrujo, Rich Farmbrough, Rschwieb, Silly rabbit, Tobias Bergemann, Vincent Semeria, Zelmerszoetrop, Zundark, 17anonymous edits

Subgroup test  Source: http://en.wikipedia.org/w/index.php?oldid=455693079  Contributors: Crazyjimbo, Geometry guy, JumpDiscont, Pt, Selfworm, Zvika, 4 anonymous edits

Subring test  Source: http://en.wikipedia.org/w/index.php?oldid=455693014  Contributors: Charles Matthews, Geometry guy, Joeldl, Oleg Alexandrov, Selfworm, Zvika, 1 anonymous edits

Sylow theorems  Source: http://en.wikipedia.org/w/index.php?oldid=455692905  Contributors: 01001, Aholtman, Alecobbe, Amitushtush, Ams80, Ank0ku, AxelBoldt, BenF, BeteNoir, CZeke,Charles Matthews, Chas zzz brown, Chochopk, Conversion script, Crisófilax, Cwkmail, David Eppstein, Derek Ross, Dominus, Druiffic, EmilJ, Eramesan, Functor salad, GTBacchus, Gauge,Geometry guy, Giftlite, Goochelaar, Graham87, Grubber, Haham hanuka, Hank hu, Headbomb, Hesam7, JackSchmidt, Japanese Searobin, Joelsims80, Jonathanzung, Kilva, Lzur, MathMartin,Mav, Michael Hardy, Nbarth, Ossido, PappyK, PierreAbbat, Pladdin, Pmanderson, Point-set topologist, Pyrop, R.e.b., Reedy, Schutz, Siroxo, Sl, Spoon!, Stove Wolf, Superninja, TakuyaMurata,Tarquin, Tobias Bergemann, Twilsonb, WLior, Welsh, Zundark, Zvika, 76 anonymous edits

Sylvester's determinant theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455692773  Contributors: Bosmon, Exol, Gentsquash, Geometry guy, Giftlite, Guillaume.bouchard, JitseNiesen, Michael Hardy, Michael Slone, Rich Farmbrough, Robinh, 9 anonymous edits

Sylvester's law of inertia  Source: http://en.wikipedia.org/w/index.php?oldid=455692673  Contributors: A. Pichler, Akriasas, Anne Bauval, Arcfrk, Bh3u4m, Can't sleep, clown will eat me,Charles Matthews, Choster, DanielJanzon, Geometry guy, Giftlite, Gryllida, Jorge Stolfi, JuJube, Michael Hardy, Nneonneo, Plclark, Randomblue, Ranicki, Seanwal111111, Shambolic Entity,Simplifix, The wub, Vanish2, 23 anonymous edits

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Takagi existence theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455768890  Contributors: Ambrose H. Field, BeteNoir, Biblbroks, Charles Matthews, Cobaltcigs, Dugwiki,Edward, Gene Ward Smith, Giftlite, R'n'B, R.e.b., Ringspectrum, Sodin, WhatamIdoing, 9 anonymous edits

Three subgroups lemma  Source: http://en.wikipedia.org/w/index.php?oldid=455692354  Contributors: Algebraist, Geometry guy, Giftlite, Michael Hardy, Omnipaedista, Point-set topologist

Trichotomy theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455693272  Contributors: Charles Matthews, Geometry guy, Gregbard, Headbomb, Michael Hardy, R.e.b., Rjwilmsi

Walter theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765520  Contributors: Captain-tucker, Gregbard, Headbomb, R.e.b., Rjwilmsi, Rschwieb

Wedderburn's little theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455693510  Contributors: CBM, Charles Matthews, Geometry guy, Giftlite, Howard McCay, Kammerer55,MarkSweep, Matikkapoika, Nbarth, Pcarpent1, Qwfp, Rgdboer, Ringspectrum, SantoBugito, TakuyaMurata, Thehotelambush, Tobias Bergemann, 5 anonymous edits

Weil conjecture on Tamagawa numbers  Source: http://en.wikipedia.org/w/index.php?oldid=455693645  Contributors: Bender235, BeteNoir, Charles Matthews, Geometry guy, Giftlite,LokiClock, R.e.b., Rjwilmsi, TakuyaMurata, 1 anonymous edits

Witt's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455693983  Contributors: Arcfrk, Blotwell, Charles Matthews, Geometry guy, Jdthomas, Michael Hardy, Nbarth, OlegAlexandrov, S11-73-3-33, Tyrrell McAllister, Ulner, 4 anonymous edits

Z* theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765703  Contributors: Giftlite, Headbomb, JackSchmidt, Jitse Niesen, Messagetolove, R.e.b., Rschwieb, Woohookitty

Zassenhaus lemma  Source: http://en.wikipedia.org/w/index.php?oldid=455718572  Contributors: Ahills60, Charles Matthews, DRLB, David Eppstein, FF2010, Giftlite, Helder.wiki, JasonRecliner, Esq., Jeepday, Julien Tuerlinckx, Mat cross, MathMartin, Michael Hardy, Nbarth, RDBury, Schneelocke, Silly rabbit, Silverfish, Waltpohl, 5 anonymous edits

ZJ theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765215  Contributors: Giftlite, JackSchmidt, Jitse Niesen, R.e.b., Rschwieb, Turgidson

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Image Sources, Licenses and ContributorsImage:Pascal's triangle 5.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Pascal's_triangle_5.svg  License: GNU Free Documentation License  Contributors: User:Conrad.Irwinoriginally User:DriniImage:Pascal triangle small.png  Source: http://en.wikipedia.org/w/index.php?title=File:Pascal_triangle_small.png  License: GNU Free Documentation License  Contributors: user:guntherImage:BinomialTheorem.png  Source: http://en.wikipedia.org/w/index.php?title=File:BinomialTheorem.png  License: Public Domain  Contributors: Danilo Guanabara FernandesImage:Birkhoff120.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Birkhoff120.svg  License: Public Domain  Contributors: David EppsteinFile:Cramer.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Cramer.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Author:Franklin Vera PachecoImage:Crystallographic restriction polygons.png  Source: http://en.wikipedia.org/w/index.php?title=File:Crystallographic_restriction_polygons.png  License: Creative CommonsAttribution-Sharealike 2.5  Contributors: -File:OEISicon light.svg  Source: http://en.wikipedia.org/w/index.php?title=File:OEISicon_light.svg  License: Public Domain  Contributors: LipediaImage:Discriminant49CubicFieldFundamentalDomainOfUnits.png  Source: http://en.wikipedia.org/w/index.php?title=File:Discriminant49CubicFieldFundamentalDomainOfUnits.png License: Creative Commons Attribution-Sharealike 3.0  Contributors: RobHarFile:Lattice diagram of Q adjoin the positive square roots of 2 and 3, its subfields, and Galois groups.svg  Source:http://en.wikipedia.org/w/index.php?title=File:Lattice_diagram_of_Q_adjoin_the_positive_square_roots_of_2_and_3,_its_subfields,_and_Galois_groups.svg  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: SelfFile:Lattice_diagram_of_Q_adjoin_a_cube_root_of_2_and_a_primitive_cube_root_of_1,_its_subfields,_and_Galois_groups.svg  Source:http://en.wikipedia.org/w/index.php?title=File:Lattice_diagram_of_Q_adjoin_a_cube_root_of_2_and_a_primitive_cube_root_of_1,_its_subfields,_and_Galois_groups.svg  License: CreativeCommons Attribution-Sharealike 3.0  Contributors: SelfImage:The four subspaces.svg  Source: http://en.wikipedia.org/w/index.php?title=File:The_four_subspaces.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors:Cronholm144image:FundHomDiag.png  Source: http://en.wikipedia.org/w/index.php?title=File:FundHomDiag.png  License: GNU Free Documentation License  Contributors: -File:Shifted square tiling.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Shifted_square_tiling.svg  License: Creative Commons Zero  Contributors: User:David EppsteinImage:First-isomorphism-theorem.svg  Source: http://en.wikipedia.org/w/index.php?title=File:First-isomorphism-theorem.svg  License: Public Domain  Contributors: Michael K. EdwardsFile:Labeled Triangle Reflections.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Labeled_Triangle_Reflections.svg  License: Public Domain  Contributors: Jim.belkFile:Hexagon Reflections.png  Source: http://en.wikipedia.org/w/index.php?title=File:Hexagon_Reflections.png  License: Public Domain  Contributors: -Image:Butterfly lemma.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Butterfly_lemma.svg  License: Creative Commons Attribution 3.0  Contributors: Claudio Rocchini

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