Characteristic (algebra)From Wikipedia, the free encyclopedia

Contents

1 Abelian extension 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Algebraic closure 22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Existence of an algebraic closure and splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Separable closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Algebraic extension 43.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Algebraic number eld 64.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Algebraicity and ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.3.1 Unique factorization and class number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3.2 -functions, L-functions and class number formula . . . . . . . . . . . . . . . . . . . . . . 8

4.4 Bases for number elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4.1 Integral basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4.2 Power basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.5 Regular representation, trace and determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.6 Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.6.1 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.6.2 Nonarchimedean or ultrametric places . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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4.6.3 Prime ideals in OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7 Ramication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.7.2 Dedekind discriminant theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.8 Galois groups and Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.9 Local-global principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.9.1 Local and global elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.9.2 Hasse principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.9.3 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Degree of a eld extension 175.1 Denition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 The multiplicativity formula for degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.1 Proof of the multiplicativity formula in the nite case . . . . . . . . . . . . . . . . . . . . 185.2.2 Proof of the formula in the innite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Dual basis in a eld extension 20

7 Field extension 217.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.5 Algebraic and transcendental elements and extensions . . . . . . . . . . . . . . . . . . . . . . . . 227.6 Normal, separable and Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.8 Extension of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 Finite eld 258.1 Denitions, rst examples, and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.3 Explicit construction of nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8.3.1 Non-prime elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

8.3.2 Field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.3.3 GF(p2) for an odd prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.3.4 GF(8) and GF(27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.3.5 GF(16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8.4 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4.1 Discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.5 Frobenius automorphism and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.6 Polynomial factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.6.1 Irreducible polynomials of a given degree . . . . . . . . . . . . . . . . . . . . . . . . . . 318.6.2 Number of monic irreducible polynomials of a given degree over a nite eld . . . . . . . . 32

8.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8.8.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.8.2 Wedderburns little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Galois extension 359.1 Characterization of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10 Normal extension 3710.1 Equivalent properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.3 Normal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11 Ring homomorphism 3911.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011.3 The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

11.3.1 Endomorphisms, isomorphisms, and automorphisms . . . . . . . . . . . . . . . . . . . . . 4111.3.2 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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11.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12 Separable extension 4212.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 4412.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4512.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4612.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

13 Simple extension 4713.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.2 Structure of simple extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

14 Splitting eld 4914.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.2 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3 Constructing splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

14.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3.2 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3.3 The eld Ki[X]/(f(X)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.4.1 The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.4.2 Cubic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

15 Tower of elds 5315.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5315.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 1

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois groupis a cyclic group, we have a cyclic extension. A Galois extension is called solvable if its Galois group is solvable, i.e.if it is constructed from a series of abelian groups corresponding to intermediate extensions.Every nite extension of a nite eld is a cyclic extension. The development of class eld theory has provided detailedinformation about abelian extensions of number elds, function elds of algebraic curves over nite elds, and localelds.There are two slightly dierent concepts of cyclotomic extensions: these can mean either extensions formed byadjoining roots of unity, or subextensions of such extensions. The cyclotomic elds are examples. Any cyclotomicextension (for either denition) is abelian.If a eld K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resultingso-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n,since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th rootsof elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-directproduct. The Kummer theory gives a complete description of the abelian extension case, and the KroneckerWebertheorem tells us that if K is the eld of rational numbers, an extension is abelian if and only if it is a subeld of a eldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classies all covering spaces of a space:abelian covers are classied by its abelianisation which relates directly to the rst homology group.

1.1 References Kuz'min, L.V. (2001), cyclotomic extension, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,

ISBN 978-1-55608-010-4

1

Chapter 2

Algebraic closure

For other uses, see Closure (disambiguation).

In mathematics, particularly abstract algebra, an algebraic closure of a eld K is an algebraic extension of K that isalgebraically closed. It is one of many closures in mathematics.Using Zorns lemma, it can be shown that every eld has an algebraic closure,[1][2][3] and that the algebraic closureof a eld K is unique up to an isomorphism that xes every member of K. Because of this essential uniqueness, weoften speak of the algebraic closure of K, rather than an algebraic closure of K.The algebraic closure of a eld K can be thought of as the largest algebraic extension of K. To see this, note that if Lis any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is containedwithin the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed eld containingK, because ifM is any algebraically closed eld containing K, then the elements ofM that are algebraic over K forman algebraic closure of K.The algebraic closure of a eld K has the same cardinality as K if K is innite, and is countably innite if K is nite.[3]

2.1 Examples The fundamental theorem of algebra states that the algebraic closure of the eld of real numbers is the eld of

complex numbers.

The algebraic closure of the eld of rational numbers is the eld of algebraic numbers.

There are many countable algebraically closed elds within the complex numbers, and strictly containing theeld of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers,e.g. the algebraic closure of Q().

For a nite eld of prime power order q, the algebraic closure is a countably innite eld that contains a copyof the eld of order qn for each positive integer n (and is in fact the union of these copies).[4]

2.2 Existence of an algebraic closure and splitting eldsLet S = ffj 2 g be the set of all monic irreducible polynomials in K[x]. For each f 2 S , introduce newvariables u;1; : : : ; u;d where d = degree(f) . Let R be the polynomial ring over K generated by u;i for all 2 and all i degree(f) . Write

f dY

i=1

(x u;i) =d1Xj=0

r;j xj 2 R[x]

2

2.3. SEPARABLE CLOSURE 3

with r;j 2 R . Let I be the ideal in R generated by the r;j . By Zorns lemma, there exists a maximal idealM in Rthat contains I. Now R/M is an algebraic closure of K; every f splits as the product of the x (u;i +M) .The same proof also shows that for any subset S of K[x], there exists a splitting eld of S over K.

2.3 Separable closureAn algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separableextensions ofK withinKalg. This subextension is called a separable closure ofK. Since a separable extension of a sep-arable extension is again separable, there are no nite separable extensions of Ksep, of degree > 1. Saying this anotherway, K is contained in a separably-closed algebraic extension eld. It is essentially unique (up to isomorphism).[5]

The separable closure is the full algebraic closure if and only if K is a perfect eld. For example, if K is a eld ofcharacteristic p and if X is transcendental over K,K(X)( p

pX) K(X) is a non-separable algebraic eld extension.

In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]

2.4 See also Algebraically closed eld Algebraic extension Puiseux expansion

2.5 References[1] McCarthy (1991) p.21

[2] M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp.11-12.

[3] Kaplansky (1972) pp.74-76

[4] Brawley, Joel V.; Schnibben, George E. (1989), 2.2 The Algebraic Closure of a Finite Field, Innite Algebraic Extensionsof Finite Fields, Contemporary Mathematics 95, American Mathematical Society, pp. 2223, ISBN 978-0-8218-5428-0,Zbl 0674.12009.

[5] McCarthy (1991) p.22

[6] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.

Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. ISBN 0-226-42451-0. Zbl 1001.16500.

McCarthy, Paul J. (1991). Algebraic extensions of elds (Corrected reprint of the 2nd ed.). New York: DoverPublications. Zbl 0768.12001.

Chapter 3

Algebraic extension

In abstract algebra, a eld extension L/K is called algebraic if every element of L is algebraic over K, i.e. if everyelement of L is a root of some non-zero polynomial with coecients in K. Field extensions that are not algebraic, i.e.which contain transcendental elements, are called transcendental.For example, the eld extensionR/Q, that is the eld of real numbers as an extension of the eld of rational numbers,is transcendental, while the eld extensionsC/R andQ(2)/Q are algebraic, whereC is the eld of complex numbers.All transcendental extensions are of innite degree. This in turn implies that all nite extensions are algebraic.[1] Theconverse is not true however: there are innite extensions which are algebraic. For instance, the eld of all algebraicnumbers is an innite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coecients in K, is not only a ring but a eld:an algebraic extension of K which has nite degree over K. The converse is true as well, if K[a] is a eld, then a isalgebraic over K. In the special case where K =Q is the eld of rational numbers, Q[a] is an example of an algebraicnumber eld.A eld with no nontrivial algebraic extensions is called algebraically closed. An example is the eld of complexnumbers. Every eld has an algebraic extension which is algebraically closed (called its algebraic closure), but provingthis in general requires some form of the axiom of choice.An extension L/K is algebraic if and only if every sub K-algebra of L is a eld.

3.1 PropertiesThe class of algebraic extensions forms a distinguished class of eld extensions, that is, the following three propertieshold:[2]

1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.

2. If E and F are algebraic extensions of K in a common overeld C, then the compositum EF is an algebraicextension of K.

3. If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K.

These nitary results can be generalized using transnite induction:

1. The union of any chain of algebraic extensions over a base eld is itself an algebraic extension over the samebase eld.

This fact, together with Zorns lemma (applied to an appropriately chosen poset), establishes the existence of algebraicclosures.

4

3.2. GENERALIZATIONS 5

3.2 GeneralizationsMain article: Substructure

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding ofM into N is calledan algebraic extension if for every x in N there is a formula p with parameters inM, such that p(x) is true and the set

ny 2 N

p(y)ois nite. It turns out that applying this denition to the theory of elds gives the usual denition of algebraic extension.The Galois group of N overM can again be dened as the group of automorphisms, and it turns out that most of thetheory of Galois groups can be developed for the general case.

3.3 See also Integral element Lroths theorem Galois extension Separable extension Normal extension

3.4 Notes[1] See also Hazewinkel et al. (2004), p. 3.

[2] Lang (2002) p.228

3.5 References Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhalovna; Kirichenko, Vladimir V. (2004),

Algebras, rings and modules 1, Springer, ISBN 1-4020-2690-0

Lang, Serge (1993), V.1:Algebraic Extensions, Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub.Co., pp. 223, ISBN 978-0-201-55540-0, Zbl 0848.13001

McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of elds, New York:Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001

Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081 Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687

Chapter 4

Algebraic number eld

In mathematics, an algebraic number eld (or simply number eld) F is a nite degree (and hence algebraic) eldextension of the eld of rational numbersQ. Thus F is a eld that containsQ and has nite dimension when consideredas a vector space over Q.The study of algebraic number elds, and, more generally, of algebraic extensions of the eld of rational numbers, isthe central topic of algebraic number theory.

4.1 Denition

4.1.1 Prerequisites

Main articles: Field and Vector space

The notion of algebraic number eld relies on the concept of a eld. A eld consists of a set of elements togetherwith two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent exampleof a eld is the eld of rational numbers, commonly denoted Q, together with its usual operations of addition etc.Another notion needed to dene algebraic number elds is vector spaces. To the extent needed here, vector spacescan be thought of as consisting of sequences (or tuples)

(x1, x2, ...)

whose entries are elements of a xed eld, such as the eld Q. Any two such sequences can be added by adding theentries one per one. Furthermore, any sequence can be multiplied by a single element c of the xed eld. These twooperations known as vector addition and scalar multiplication satisfy a number of properties that serve to dene vectorspaces abstractly. Vector spaces are allowed to be innite-dimensional, that is to say that the sequences constitutingthe vector spaces are of innite length. If, however, the vector space consists of nite sequences

(x1, x2, ..., xn),

the vector space is said to be of nite dimension, n.

4.1.2 Denition

An algebraic number eld (or simply number eld) is a nite degree eld extension of the eld of rational numbers.Here its dimension as a vector space over Q is simply called its degree.

6

4.2. EXAMPLES 7

4.2 Examples The smallest and most basic number eld is the eldQ of rational numbers. Many properties of general number

elds, such as unique factorization, are modelled after the properties of Q.

The Gaussian rationals, denoted Q(i) (read as "Q adjoined i"), form the rst nontrivial example of a numbereld. Its elements are expressions of the form

a+bi

where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added,subtracted, andmultiplied according to the usual rules of arithmetic and then simplied using the identity

i2 = 1.

Explicitly,

(a + bi) + (c + di) = (a + c) + (b + d)i,(a + bi) (c + di) = (ac bd) + (ad + bc)i.

Non-zero Gaussian rational numbers are invertible, which can be seen from the identity

(a+ bi)

a

a2 + b2 ba2 + b2

i

=

(a+ bi)(a bi)a2 + b2

= 1:

It follows that the Gaussian rationals form a number eld which is two-dimensional as a vector spaceover Q.

More generally, for any square-free integer d, the quadratic eld

Q(d)

is a number eld obtained by adjoining the square root of d to the eld of rational numbers. Arithmeticoperations in this eld are dened in analogy with the case of gaussian rational numbers, d = 1.

Cyclotomic eld

Q(n), n = exp (2i / n)

is a number eld obtained from Q by adjoining a primitive nth root of unity n. This eld containsall complex nth roots of unity and its dimension over Q is equal to (n), where is the Euler totientfunction.

The real numbers,R, and the complex numbers,C, are elds which have innite dimension asQ-vector spaces,hence, they are not number elds. This follows from the uncountability of R and C as sets, whereas everynumber eld is necessarily countable.

The set Q2 of ordered pairs of rational numbers, with the entrywise addition and multiplication is a two-dimensional commutative algebra over Q. However, it is not a eld, since it has zero divisors:

(1, 0) (0, 1) = (1 0, 0 1) = (0, 0).

8 CHAPTER 4. ALGEBRAIC NUMBER FIELD

4.3 Algebraicity and ring of integersGenerally, in abstract algebra, a eld extension F / E is algebraic if every element f of the bigger eld F is the zeroof a polynomial with coecients e0, ..., em in E:

p(f) = emfm + emfm1 + ... + e1f + e0 = 0.

It is a fact that every eld extension of nite degree is algebraic (proof: for x in F simply consider 1, x, x2, x3, ..., weget a linear dependence, i.e. a polynomial that x is a root of!) because of the nite degree. In particular this appliesto algebraic number elds, so any element f of an algebraic number eld F can be written as a zero of a polynomialwith rational coecients. Therefore, elements of F are also referred to as algebraic numbers. Given a polynomialp such that p(f) = 0, it can be arranged such that the leading coecient em is one, by dividing all coecients byit, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rationalcoecients. If, however, its coecients are actually all integers, f is called an algebraic integer. Any (usual) integerz Z is an algebraic integer, as it is the zero of the linear monic polynomial:

p(t) = t z.

It can be shown that any algebraic integer that is also a rational number must actually be an integer, whence thename algebraic integer. Again using abstract algebra, specically the notion of a nitely generated module, it canbe shown that the sum and the product of any two algebraic integers is still an algebraic integer, it follows that thealgebraic integers in F form a ring denoted OF called the ring of integers of F. It is a subring of (that is, a ringcontained in) F. A eld contains no zero divisors and this property is inherited by any subring. Therefore, the ringof integers of F is an integral domain. The eld F is the eld of fractions of the integral domain OF. This way onecan get back and forth between the algebraic number eld F and its ring of integers OF. Rings of algebraic integershave three distinctive properties: rstly, OF is an integral domain that is integrally closed in its eld of fractions F.Secondly, OF is a Noetherian ring. Finally, every nonzero prime ideal of OF is maximal or, equivalently, the Krulldimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (orDedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.

4.3.1 Unique factorization and class numberFor general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product ofprime ideals. However, unlike Z as the ring of integers of Q, the ring of integers of a proper extension of Q neednot admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. Thishappens already for quadratic integers, for example in OQ = Z[5], the uniqueness of the factorization fails:

6 = 2 3 = (1 + 5) (1 5).

Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors donot just dier by a unit in OQ. Euclidean domains are unique factorization domains; for example Z[i], the ringof Gaussian integers, and Z[], the ring of Eisenstein integers, where is a third root of unity (unequal to 1), havethis property.[1]

4.3.2 -functions, L-functions and class number formulaThe failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of theso-called ideal class group. This group is always nite. The ring of integers OF possesses unique factorization if andonly if it is a principal ring or, equivalently, if F has class number 1. Given a number eld, the class number is oftendicult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginaryquadratic number elds (i.e., Q(d), d 1) with prescribed class number. The class number formula relates h toother fundamental invariants of F. It involves the Dedekind zeta function F(s), a function in a complex variable s,dened by

F (s) :=Yp

1

1N(p)s

4.4. BASES FOR NUMBER FIELDS 9

(The product is over all prime ideals of OF, N(p) denotes the norm of the prime ideal or, equivalently, the (-nite) number of elements in the residue eld OF /p . The innite product converges only for Re(s) > 1, in generalanalytic continuation and the functional equation for the zeta-function are needed to dene the function for all s). TheDedekind zeta-function generalizes the Riemann zeta-function in that Q(s) = (s).The class number formula states that F(s) has a simple pole at s = 1 and at this point (its meromorphic continuationto the whole complex plane) the residue is given by

2r1 (2)r2 h Regw pjDj :

Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of F, respectively.Moreover, Reg is the regulator of F, w the number of roots of unity in F and D is the discriminant of F.Dirichlet L-functions L(, s) are a more rened variant of (s). Both types of functions encode the arithmetic behaviorof Q and F, respectively. For example, Dirichlets theorem asserts that in any arithmetic progression

a, a + m, a + 2m, ...

with coprime a andm, there are innitely many prime numbers. This theorem is implied by the fact that the DirichletL-function is nonzero at s = 1. Using much more advanced techniques including algebraic K-theory and Tamagawameasures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture),of values of more general L-functions.[2]

4.4 Bases for number elds

4.4.1 Integral basisAn integral basis for a number eld F of degree n is a set

B = {b1, , bn}

of n algebraic integers in F such that every element of the ring of integers OF of F can be written uniquely as aZ-linear combination of elements of B; that is, for any x in OF we have

x = m1b1 + + mnbn,

where the mi are (ordinary) integers. It is then also the case that any element of F can be written uniquely as

m1b1 + + mnbn,

where now the mi are rational numbers. The algebraic integers of F are then precisely those elements of F where themi are all integers.Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis,and it is now standard for computer algebra systems to have built-in programs to do this.

4.4.2 Power basisLet F be a number eld of degree n. Among all possible bases of F (seen as a Q-vector space), there are particularones known as power bases, that are bases of the form

Bx = {1, x, x2, ..., xn1}

for some element x F. By the primitive element theorem, there exists such an x, called a primitive element. If x canbe chosen in OF and such that Bx is a basis of OF as a free Z-module, then Bx is called a power integral basis, and theeld F is called a monogenic eld. An example of a number eld that is not monogenic was rst given by Dedekind.His example is the eld obtained by adjoining a root of the polynomial x3 x2 2x 8.[3]

10 CHAPTER 4. ALGEBRAIC NUMBER FIELD

4.5 Regular representation, trace and determinant

Using the multiplication in F, the elements of the eld F may be represented by n-by-n matrices

A = A(x)=(aij) i, j n,

by requiring

xei =

nXj=1

aijej ; aij 2 Q:

Here e1, ..., en is a xed basis for F, viewed as a Q-vector space. The rational numbers aij are uniquely determinedby x and the choice of a basis since any element of F can be uniquely represented as a linear combination of the basiselements. This way of associating a matrix to any element of the eld F is called the regular representation. Thesquare matrix A represents the eect of multiplication by x in the given basis. It follows that if the element y of F isrepresented by a matrix B, then the product xy is represented by the matrix product BA. Invariants of matrices, suchas the trace, determinant, and characteristic polynomial, depend solely on the eld element x and not on the basis. Inparticular, the trace of the matrix A(x) is called the trace of the eld element x and denoted Tr(x), and the determinantis called the norm of x and denoted N(x).By denition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linearfunction of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(x) = Tr(x), and the norm is amultiplicative homogeneousfunction of degree n: N(xy) = N(x) N(y), N(x) = n N(x). Here is a rational number, and x, y are any two elementsof F.The trace form derives is a bilinear form dened by means of the trace, as Tr(x y). The integral trace form, an integer-valued symmetric matrix is dened as t = Tr(bb), where b1, ..., b is an integral basis for F. The discriminant of Fis dened as det(t). It is an integer, and is an invariant property of the eld F, not depending on the choice of integralbasis.The matrix associated to an element x of F can also be used to give other, equivalent descriptions of algebraic integers.An element x of F is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated tox is a monic polynomial with integer coecients. Suppose that the matrix A that represents an element x has integerentries in some basis e. By the CayleyHamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is analgebraic integer. Conversely, if x is an element of F which is a root of a monic polynomial with integer coecientsthen the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrixin a suitable basis of F. Note that the property of being an algebraic integer is dened in a way that is independent ofa choice of a basis in F.

4.5.1 Example

Consider F = Q(x), where x satises x3 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and thecorresponding integral trace form is

24 3 11 6111 119 65361 653 3589

35:The 3 in the upper left hand corner of this matrix is the trace of the matrix of the map dened by the rst basiselement (1) in the regular representation of F on F. This basis element induces the identity map on the 3-dimensionalvector space, F. The trace of the matrix of the identity map on a 3-dimensional vector space is 3.The determinant of this is 1304 = 23163, the eld discriminant; in comparison the root discriminant, or discriminantof the polynomial, is 5216 = 25163.

4.6. PLACES 11

4.6 PlacesMathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.[4][5] Thissituation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all ofthe various possible embeddings of a number eld F into its various topological completions at once.A place of a number eld F is an equivalence class of absolute values on F. Essentially, an absolute value is a notion tomeasure the size of elements f of F. Two such absolute values are considered equivalent if they give rise to the samenotion of smallness (or proximity). In general, they fall into three regimes. Firstly (and mostly irrelevant), the trivialabsolute value | |0, which takes the value 1 on all non-zero f in F. The second and third classes are Archimedeanplaces and non-Archimedean (or ultrametric) places. The completion of F with respect to a place is given in bothcases by taking Cauchy sequences in F and dividing out null sequences, that is, sequences (xn)n N such that |xn|tends to zero when n tends to innity. This can be shown to be a eld again, the so-called completion of F at thegiven place.For F = Q, the following non-trivial norms occur (Ostrowskis theorem): the (usual) absolute value, which givesrise to the complete topological eld of the real numbers R. On the other hand, for any prime number p, the p-adicabsolute values is dened by

|q|p = pn, where q = pn a/b and a and b are integers not divisible by p.

In contrast to the usual absolute value, the p-adic norm gets smaller when q is multiplied by p, leading to quite dierentbehavior of Qp vis--vis R.

4.6.1 Archimedean places[6][7]

For some of the details take a look at,[8] Chapter 11 C p. 108. Note in particular the standard notation r1 and r2 forthe number of real and complex embeddings, respectively (see below).Calculating the archimedean places of F is done as follows: let x be a primitive element of F, with minimal polynomial(over Q) f. Over R, f will generally no longer be irreducible, but its irreducible (real) factors are either of degreeone or two. Since there are no repeated roots, there are no repeated factors. The roots r of factors of degree one arenecessarily real, and replacing x by r gives an embedding of F into R; the number of such embeddings is equal to thenumber of real roots of f. Restricting the standard absolute value on R to F gives an archimedean absolute value onF; such an absolute value is also referred to as a real place of F. On the other hand, the roots of factors of degree twoare pairs of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pairof embeddings can be used to dene an absolute value on F, which is the same for both embeddings since they areconjugate. This absolute value is called a complex place of F.If all roots of f above are real (respectively, complex) or, equivalently, any possible embedding F C is actuallyforced to be inside R (resp. C), F is called totally real (resp. totally complex).

4.6.2 Nonarchimedean or ultrametric places

To nd the nonarchimedean places, let again f and x be as above. In Qp, f splits in factors of various degrees, noneof which are repeated, and the degrees of which add up to n, the degree of f. For each of these p-adically irreduciblefactors t, we may suppose that x satises t and obtain an embedding of F into an algebraic extension of nite degreeover Q. Such a local eld behaves in many ways like a number eld, and the p-adic numbers may similarly play therole of the rationals; in particular, we can dene the norm and trace in exactly the same way, now giving functionsmapping toQp. By using this p-adic norm mapNt for the place t, we may dene an absolute value corresponding to agiven p-adically irreducible factor t of degree m by ||t = |Nt()|p1/m. Such an absolute value is called an ultrametric,non-Archimedean or p-adic place of F.For any ultrametric place v we have that |x|v 1 for any x in OF, since the minimal polynomial for x has integerfactors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for eachfactor is a p-adic integer, and one of these is the integer used for dening the absolute value for v.

12 CHAPTER 4. ALGEBRAIC NUMBER FIELD

4.6.3 Prime ideals in OFFor an ultrametric place v, the subset of OF dened by |x|v < 1 is an ideal P of OF. This relies on the ultrametricityof v: given x and y in P, then

|x + y|v max (|x|v, |y|v) < 1.

Actually, P is even a prime ideal.Conversely, given a prime ideal P of OF, a discrete valuation can be dened by setting vP(x) = n where n is thebiggest integer such that x Pn, the n-fold power of the ideal. This valuation can be turned into an ultrametric place.Under this correspondence, (equivalence classes) of ultrametric places of F correspond to prime ideals of OF. ForF = Q, this gives back Ostrowskis theorem: any prime ideal in Z (which is necessarily by a single prime number)corresponds to an non-archimedean place and vice versa. However, for more general number elds, the situationbecomes more involved, as will be explained below.Yet another, equivalent way of describing ultrametric places is by means of localizations of OF. Given an ultrametricplace v on a number eld F, the corresponding localization is the subring T of F of all elements x such that | x |v 1.By the ultrametric property T is a ring. Moreover, it contains OF. For every element x of F, at least one of x or x1is contained in T. Actually, since F/T can be shown to be isomorphic to the integers, T is a discrete valuation ring,in particular a local ring. Actually, T is just the localization of OF at the prime ideal P. Conversely, P is the maximalideal of T.Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations ona number eld.

4.7 Ramication

Schematic depiction of ramication: the bers of almost all points in Y below consist of three points, except for two points in Ymarked with dots, where the bers consist of one and two points (marked in black), respectively. The map f is said to be ramied inthese points of Y.

Ramication, generally speaking, describes a geometric phenomenon that can occur with nite-to-one maps (that is,maps f: X Y such that the preimages of all points y in Y consist only of nitely many points): the cardinality ofthe bers f1(y) will generally have the same number of points, but it occurs that, in special points y, this numberdrops. For example, the map

C C, z zn

has n points in each ber over t, namely the n (complex) roots of t, except in t = 0, where the ber consists of only oneelement, z = 0. One says that the map is ramied in zero. This is an example of a branched covering of Riemannsurfaces. This intuition also serves to dene ramication in algebraic number theory. Given a (necessarily nite)extension of number elds F / E, a prime ideal p of OE generates the ideal pOF of OF. This ideal may or may not bea prime ideal, but, according to the LaskerNoether theorem (see above), always is given by

4.8. GALOIS GROUPS AND GALOIS COHOMOLOGY 13

pOF = q1e1 q2e2 ... qmem

with uniquely determined prime ideals qi of OF and numbers (called ramication indices) ei. Whenever one rami-cation index is bigger than one, the prime p is said to ramify in F.The connection between this denition and the geometric situation is delivered by the map of spectra of rings SpecOFSpecOE. In fact, unramied morphisms of schemes in algebraic geometry are a direct generalization of unramiedextensions of number elds.Ramication is a purely local property, i.e., depends only on the completions around the primes p and qi. The inertiagroup measures the dierence between the local Galois groups at some place and the Galois groups of the involvednite residue elds.

4.7.1 An exampleThe following example illustrates the notions introduced above. In order to compute the ramication index of Q(x),where

f(x) = x3 x 1 = 0,

at 23, it suces to consider the eld extensionQ23(x) /Q23. Up to 529 = 232 (i.e., modulo 529) f can be factored as

f(x) = (x + 181)(x2 181x 38) = gh.

Substituting x = y + 10 in the rst factor gmodulo 529 yields y + 191, so the valuation | y |g for y given by g is | 191|23 = 1. On the other hand the same substitution in h yields y2 161y 161 modulo 529. Since 161 = 7 23,

|y|h = 16123 = 1 / 23.

Since possible values for the absolute value of the place dened by the factor h are not conned to integer powers of23, but instead are integer powers of the square root of 23, the ramication index of the eld extension at 23 is two.The valuations of any element of F can be computed in this way using resultants. If, for example y = x2 x 1, usingthe resultant to eliminate x between this relationship and f = x3 x 1 = 0 gives y3 5y2 + 4y 1 = 0. If insteadwe eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y,and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g andh (which are both 1 in this instance.)

4.7.2 Dedekind discriminant theoremMuch of the signicance of the discriminant lies in the fact that ramied ultrametric places are all places obtainedfrom factorizations inQpwhere p divides the discriminant. This is even true of the polynomial discriminant; howeverthe converse is also true, that if a prime p divides the discriminant, then there is a p-place which ramies. For thisconverse the eld discriminant is needed. This is the Dedekind discriminant theorem. In the example above, thediscriminant of the number eld Q(x) with x3 x 1 = 0 is 23, and as we have seen the 23-adic place ramies.The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramied place comes fromthe absolute value on the complex embedding of F.

4.8 Galois groups and Galois cohomologyGenerally in abstract algebra, eld extensionsF /E can be studied by examining theGalois groupGal(F /E), consistingof eld automorphisms of F leaving E elementwise xed. As an example, the Galois group Gal (Q(n) / Q) of thecyclotomic eld extension of degree n (see above) is given by (Z/nZ), the group of invertible elements in Z/nZ. Thisis the rst stepstone into Iwasawa theory.In order to include all possible extensions having certain properties, the Galois group concept is commonly appliedto the (innite) eld extension F / F of the algebraic closure, leading to the absolute Galois group G := Gal(F / F)

14 CHAPTER 4. ALGEBRAIC NUMBER FIELD

or just Gal(F), and to the extension F / Q. The fundamental theorem of Galois theory links elds in between F andits algebraic closure and closed subgroups of Gal (F). For example, the abelianization (the biggest abelian quotient)Gab of G corresponds to a eld referred to as the maximal abelian extension Fab (called so since any further extensionis not abelian, i.e., does not have an abelian Galois group). By the KroneckerWeber theorem, the maximal abelianextension of Q is the extension generated by all roots of unity. For more general number elds, class eld theory,specically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group. Also notableis the Hilbert class eld, the maximal abelian unramied eld extension of F. It can be shown to be nite over F, itsGalois group over F is isomorphic to the class group of F, in particular its degree equals the class number h of F (seeabove).In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is thenalso referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal(F), alsoknown as Galois cohomology, which in the rst place measures the failure of exactness of taking Gal(F)-invariants,but oers deeper insights (and questions) as well. For example, the Galois group G of a eld extension L / F actson L, the nonzero elements of L. This Galois module plays a signicant role in many arithmetic dualities, such asPoitou-Tate duality. The Brauer group of F, originally conceived to classify division algebras over F, can be recast asa cohomology group, namely H2(Gal (F), F).

4.9 Local-global principle

Generally speaking, the term local to global refers to the idea that a global problem is rst done at a local level,which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be puttogether to get back to some global statement. For example, the notion of sheaves reies that idea in topology andgeometry.

4.9.1 Local and global elds

Number elds share a great deal of similarity with another class of elds much used in algebraic geometry known asfunction elds of algebraic curves over nite elds. An example is Fp(T). They are similar in many respects, for ex-ample in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient elds of whichis the function eld in question) of curves. Therefore, both types of eld are called global elds. In accordance withthe philosophy laid out above, they can be studied at a local level rst, that is to say, by looking at the correspondinglocal elds. For number elds F, the local elds are the completions of F at all places, including the archimedeanones (see local analysis). For function elds, the local elds are completions of the local rings at all points of thecurve for function elds.Many results valid for function elds also hold, at least if reformulated properly, for number elds. However, thestudy of number elds often poses diculties and phenomena not encountered in function elds. For example, infunction elds, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function eldsoften serves as a source of intuition what should be expected in the number eld case.

4.9.2 Hasse principle

A prototypical question, posed at a global level, is whether some polynomial equation has a solution in F. If thisis the case, this solution is also a solution in all completions. The local-global principle or Hasse principle assertsthat for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solutioncan be done on all the completions of F, which is often easier, since analytic methods (classical analytic tools suchas intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can beused. This implication does not hold, however, for more general types of equations. However, the idea of passingfrom local data to global ones proves fruitful in class eld theory, for example, where local class eld theory is usedto obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completionsF can be explicitly determined, whereas the Galois groups of global elds, even of Q are far less understood.

4.10. SEE ALSO 15

4.9.3 Adeles and idelesIn order to assemble local data pertaining to all local elds attached to F, the adele ring is set up. A multiplicativevariant is referred to as ideles.

4.10 See also Dirichlets unit theorem, S-unit Kummer extension Minkowskis theorem, Geometry of numbers Chebotarevs density theorem Ray class group Decomposition group Genus eld

4.11 Notes[1] Ireland, Kenneth; Rosen, Michael (1998), A Classical Introduction to Modern Number Theory, Berlin, New York: Springer-

Verlag, ISBN 978-0-387-97329-6, Ch. 1.4

[2] Bloch, Spencer; Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol.I, Progr. Math. 86, Boston, MA: Birkhuser Boston, pp. 333400, MR 1086888

[3] Narkiewicz 2004, 2.2.6

[4] Kleiner, Israel (1999), Field theory: from equations to axiomatization. I, The American Mathematical Monthly 106 (7):677684, doi:10.2307/2589500, MR 1720431, To Dedekind, then, elds were subsets of the complex numbers.

[5] Mac Lane, Saunders (1981), Mathematical models: a sketch for the philosophy of mathematics, The American Mathe-matical Monthly 88 (7): 462472, doi:10.2307/2321751, MR 628015, Empiricism sprang from the 19th-century view ofmathematics as almost coterminal with theoretical physics.

[6] Cohn

[7] Conrad

[8] Cohn

4.12 References Cohn, Harvey (1988), A Classical Invitation to Algebraic Numbers and Class Fields, Universitext, New York:

Springer-Verlag

Conrad, Keith http://www.math.uconn.edu/~{}kconrad/blurbs/gradnumthy/unittheorem.pdf Janusz, Gerald J. (1996), Algebraic Number Fields (2nd ed.), Providence, R.I.: American Mathematical Soci-

ety, ISBN 978-0-8218-0429-2

Helmut Hasse, Number Theory, Springer Classics in Mathematics Series (2002) Serge Lang, Algebraic Number Theory, second edition, Springer, 2000 Richard A. Mollin, Algebraic Number Theory, CRC, 1999 Ram Murty, Problems in Algebraic Number Theory, Second Edition, Springer, 2005

16 CHAPTER 4. ALGEBRAIC NUMBER FIELD

Narkiewicz, Wadysaw (2004), Elementary and analytic theory of algebraic numbers, Springer Monographsin Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267

Neukirch, Jrgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322,Berlin, New York: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021

Neukirch, Jrgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehrender MathematischenWissenschaften 323, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196, Zbl 1136.11001

Andr Weil, Basic Number Theory, third edition, Springer, 1995

Chapter 5

Degree of a eld extension

In mathematics, more specically eld theory, the degree of a eld extension is a rough measure of the size ofthe eld extension. The concept plays an important role in many parts of mathematics, including algebra and numbertheory indeed in any area where elds appear prominently.

5.1 Denition and notation

Suppose that E/F is a eld extension. Then E may be considered as a vector space over F (the eld of scalars). Thedimension of this vector space is called the degree of the eld extension, and it is denoted by [E:F].The degree may be nite or innite, the eld being called a nite extension or innite extension accordingly. Anextension E/F is also sometimes said to be simply nite if it is a nite extension; this should not be confused with theelds themselves being nite elds (elds with nitely many elements).The degree should not be confused with the transcendence degree of a eld; for example, the eld Q(X) of rationalfunctions has innite degree over Q, but transcendence degree only equal to 1.

5.2 The multiplicativity formula for degrees

Given three elds arranged in a tower, say K a subeld of L which is in turn a subeld ofM, there is a simple relationbetween the degrees of the three extensions L/K, M/L and M/K:

[M : K] = [M : L] [L : K]:

In other words, the degree going from the bottom to the top eld is just the product of the degrees going fromthe bottom to the middle and then from the middle to the top. It is quite analogous to Lagranges theorem ingroup theory, which relates the order of a group to the order and index of a subgroup indeed Galois theory showsthat this analogy is more than just a coincidence.The formula holds for both nite and innite degree extensions. In the innite case, the product is interpreted in thesense of products of cardinal numbers. In particular, this means that if M/K is nite, then both M/L and L/K arenite.IfM/K is nite, then the formula imposes strong restrictions on the kinds of elds that can occur betweenM andK, viasimple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediateeld L, one of two things can happen: either [M:L] = p and [L:K] = 1, in which case L is equal to K, or [M:L] = 1 and[L:K] = p, in which case L is equal toM. Therefore there are no intermediate elds (apart fromM and K themselves).

17

18 CHAPTER 5. DEGREE OF A FIELD EXTENSION

5.2.1 Proof of the multiplicativity formula in the nite caseSuppose that K, L andM form a tower of elds as in the degree formula above, and that both d = [L:K] and e = [M:L]are nite. This means that we may select a basis {u1, ..., ud} for L over K, and a basis {w1, ..., we} forM over L. Wewill show that the elements umwn, for m ranging through 1, 2, ..., d and n ranging through 1, 2, ..., e, form a basis forM/K; since there are precisely de of them, this proves that the dimension of M/K is de, which is the desired result.First we check that they spanM/K. If x is any element ofM, then since the wn form a basis forM over L, we can ndelements an in L such that

x =eX

n=1

anwn = a1w1 + + aewe:

Then, since the um form a basis for L over K, we can nd elements bm,n in K such that for each n,

an =dX

m=1

bm;num = b1;nu1 + + bd;nud:

Then using the distributive law and associativity of multiplication in M we have

x =eX

n=1

dX

m=1

bm;num

!wn =

eXn=1

dXm=1

bm;n(umwn);

which shows that x is a linear combination of the umwn with coecients from K; in other words they spanM over K.Secondly we must check that they are linearly independent over K. So assume that

0 =eX

n=1

dXm=1

bm;n(umwn)

for some coecients bm,n in K. Using distributivity and associativity again, we can group the terms as

0 =eX

n=1

dX

m=1

bm;num

!wn;

and we see that the terms in parentheses must be zero, because they are elements of L, and the wn are linearlyindependent over L. That is,

0 =dX

m=1

bm;num

for each n. Then, since the bm,n coecients are in K, and the um are linearly independent over K, we must have thatbm,n = 0 for all m and all n. This shows that the elements umwn are linearly independent over K. This concludes theproof.

5.2.2 Proof of the formula in the innite caseIn this case, we start with bases u and w of L/K and M/L respectively, where is taken from an indexing set A,and from an indexing set B. Using an entirely similar argument as the one above, we nd that the products uwform a basis for M/K. These are indexed by the cartesian product A B, which by denition has cardinality equal tothe product of the cardinalities of A and B.

5.3. EXAMPLES 19

5.3 Examples The complex numbers are a eld extension over the real numbers with degree [C:R] = 2, and thus there are no

non-trivial elds between them. The eld extension Q(2, 3), obtained by adjoining 2 and 3 to the eld Q of rational numbers, has degree

4, that is, [Q(2, 3):Q] = 4. The intermediate eld Q(2) has degree 2 over Q; we conclude from themultiplicativity formula that [Q(2, 3):Q(2)] = 4/2 = 2.

The nite eld GF(125) = GF(53) has degree 3 over its subeld GF(5). More generally, if p is a prime and n,m are positive integers with n dividing m, then [GF(pm):GF(pn)] = m/n.

The eld extensionC(T)/C, whereC(T) is the eld of rational functions overC, has innite degree (indeed it isa purely transcendental extension). This can be seen by observing that the elements 1, T, T2, etc., are linearlyindependent over C.

The eld extension C(T2) also has innite degree over C. However, if we view C(T2) as a subeld of C(T),then in fact [C(T):C(T2)] = 2. More generally, if X and Y are algebraic curves over a eld K, and F : X Yis a surjective morphism between them of degree d, then the function elds K(X) and K(Y) are both of innitedegree over K, but the degree [K(X):K(Y)] turns out to be equal to d.

5.4 GeneralizationGiven two division rings E and F with F contained in E and the multiplication and addition of F being the restrictionof the operations in E, we can consider E as a vector space over F in two ways: having the scalars act on the left,giving a dimension [E:F], and having them act on the right, giving a dimension [E:F]. The two dimensions need notagree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above appliesto left-acting scalars without change.

5.5 References page 215, Jacobson, N. (1985). Basic Algebra I. W. H. Freeman and Company. ISBN 0-7167-1480-9. Proof

of the multiplicativity formula. page 465, Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. ISBN 0-7167-1933-9. Briey

discusses the innite dimensional case.

Chapter 6

Dual basis in a eld extension

In mathematics, the linear algebra concept of dual basis can be applied in the context of a nite extension L/K, byusing the eld trace. This requires the property that the eld trace TrL/K provides a non-degenerate quadratic formover K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect eld, and hencein the cases where K is nite, or of characteristic zero.A dual basis is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using asecond basis for computations.Consider two bases for elements in a nite eld, GF(pm):

B1 = 0; 1; : : : ; m1

and

B2 = 0; 1; : : : ; m1

then B2 can be considered a dual basis of B1 provided

Tr(i j) =0; if i 6= j1; otherwise

Here the trace of a value in GF(pm) can be calculated as follows:

Tr() =m1Xi=0

pi

Using a dual basis can provide a way to easily communicate between devices that use dierent bases, rather than havingto explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implementedthen conversion from an element in the original basis to the dual basis can be accomplished with a multiplication bythe multiplicative identity (usually 1).

20

Chapter 7

Field extension

In abstract algebra, eld extensions are the main object of study in eld theory. The general idea is to start with abase eld and construct in some manner a larger eld that contains the base eld and satises additional properties.For instance, the set Q(2) = {a + b2 | a, b Q} is the smallest extension of Q that includes every real solution tothe equation x2 = 2.

7.1 Denitions

Let L be a eld. A subeld of L is a subset K of L that is closed under the eld operations of L and under takinginverses in L. In other words, K is a eld with respect to the eld operations inherited from L. The larger eld L isthen said to be an extension eld of K. To simplify notation and terminology, one says that L / K (read as "L overK") is a eld extension to signify that L is an extension eld of K.If L is an extension of F which is in turn an extension ofK, then F is said to be an intermediate eld (or intermediateextension or subextension) of the eld extension L /K.Given a eld extension L /K and a subset S of L, the smallest subeld of L which contains K and S is denoted byK(S)i.e. K(S) is the eld generated by adjoining the elements of S to K. If S consists of only one element s, K(s) isa shorthand for K({s}). A eld extension of the form L = K(s) is called a simple extension and s is called a primitiveelement of the extension.Given a eld extension L /K, the larger eld L can be considered as a vector space over K. The elements of L arethe vectors and the elements of K are the scalars, with vector addition and scalar multiplication obtained fromthe corresponding eld operations. The dimension of this vector space is called the degree of the extension and isdenoted by [L : K].An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is nite orinnite the extension is called a nite extension or innite extension.

7.2 Caveats

The notation L /K is purely formal and does not imply the formation of a quotient ring or quotient group or any otherkind of division. Instead the slash expresses the word over. In some literature the notation L:K is used.It is often desirable to talk about eld extensions in situations where the small eld is not actually contained in thelarger one, but is naturally embedded. For this purpose, one abstractly denes a eld extension as an injective ringhomomorphism between two elds. Every non-zero ring homomorphism between elds is injective because elds donot possess nontrivial proper ideals, so eld extensions are precisely the morphisms in the category of elds.Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subelds.

21

22 CHAPTER 7. FIELD EXTENSION

7.3 ExamplesThe eld of complex numbers C is an extension eld of the eld of real numbers R, and R in turn is an extensioneld of the eld of rational numbersQ. Clearly then, C/Q is also a eld extension. We have [C : R] = 2 because {1,i}is a basis, so the extension C/R is nite. This is a simple extension because C=R( i ). [R : Q] = c (the cardinality ofthe continuum), so this extension is innite.The set Q(2) = {a + b2 | a, b Q} is an extension eld of Q, also clearly a simple extension. The degree is 2because {1, 2} can serve as a basis. Q(2, 3) = Q(2)( 3)={a + b3 | a, b Q(2)}={a + b2+ c3+ d6 | a,b,c,d Q} is an extension eld of both Q(2) and Q, of degree 2 and 4 respectively. Finite extensions of Q are alsocalled algebraic number elds and are important in number theory.Another extension eld of the rationals, quite dierent in avor, is the eld of p-adic numbersQp for a prime numberp.It is common to construct an extension eld of a given eld K as a quotient ring of the polynomial ring K[X] in orderto create a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 =1. Then the polynomial X2 + 1 is irreducible in K[X], consequently the ideal (X2 + 1) generated by this polynomialis maximal, and L = K[X]/(X2 + 1) is an extension eld of K which does contain an element whose square is 1(namely the residue class of X).By iterating the above construction, one can construct a splitting eld of any polynomial from K[X]. This is anextension eld L of K in which the given polynomial splits into a product of linear factors.If p is any prime number and n is a positive integer, we have a nite eld GF(pn) with pn elements; this is an extensioneld of the nite eld GF(p) = Z/pZ with p elements.Given a eld K, we can consider the eld K(X) of all rational functions in the variable X with coecients in K; theelements of K(X) are fractions of two polynomials over K, and indeed K(X) is the eld of fractions of the polynomialring K[X]. This eld of rational functions is an extension eld of K. This extension is innite.Given a Riemann surface M, the set of all meromorphic functions dened on M is a eld, denoted by C(M). It is anextension eld of C, if we identify every complex number with the corresponding constant function dened on M.Given an algebraic varietyV over some eldK, then the function eld ofV, consisting of the rational functions denedon V and denoted by K(V), is an extension eld of K.

7.4 Elementary propertiesIf L/K is a eld extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup of(L,+), and the multiplicative group (K{0},) is a subgroup of (L{0},). In particular, if x is an element of K, thenits additive inverse x computed in K is the same as the additive inverse of x computed in L; the same is true formultiplicative inverses of non-zero elements of K.In particular then, the characteristics of L and K are the same.

7.5 Algebraic and transcendental elements and extensionsIf L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraicover K. Elements that are not algebraic are called transcendental. For example:

In C/R, i is algebraic because it is a root of x2 + 1. In R/Q, 2 + 3 is algebraic, because it is a root[1] of x4 10x2 + 1 In R/Q, e is transcendental because there is no polynomial with rational coecients that has e as a root (see

transcendental number) In C/R, e is algebraic because it is the root of x e

The special case ofC/Q is especially important, and the names algebraic number and transcendental number are usedto describe the complex numbers that are algebraic and transcendental (respectively) over Q.

7.6. NORMAL, SEPARABLE AND GALOIS EXTENSIONS 23

If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it issaid to be a transcendental extension.A subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coecients in Kexists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendencedegree of L/K. It is always possible to nd a set S, algebraically independent over K, such that L/K(S) is algebraic.Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to thetranscendence degree of the extension. An extension L/K is said to be purely transcendental if and only if thereexists a transcendence basis S of L/K such that L=K(S). Such an extension has the property that all elements of Lexcept those of K are transcendental over K, but, however, there are extensions with this property which are notpurely transcendentala class of such extensions take the form L/K where both L and K are algebraically closed.In addition, if L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarilyfollow that L=K(S). (For example, consider the extension Q(x,x)/Q, where x is transcendental over Q. The set {x}is algebraically independent since x is transcendental. Obviously, the extension Q(x,x)/Q(x) is algebraic, hence {x}is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x forx. But it is easy to see that {x} is a transcendence basis that generates Q(x,x)), so this extension is indeed purelytranscendental.)It can be shown that an extension is algebraic if and only if it is the union of its nite subextensions. In particular,every nite extension is algebraic. For example,

C/R and Q(2)/Q, being nite, are algebraic.

R/Q is transcendental, although not purely transcendental.

K(X)/K is purely transcendental.

A simple extension is nite if generated by an algebraic element, and purely transcendental if generated by a tran-scendental element. So

R/Q is not simple, as it is neither nite nor purely transcendental.

Every eld K has an algebraic closure; this is essentially the largest extension eld of K which is algebraic over K andwhich contains all roots of all polynomial equations with coecients in K. For example, C is the algebraic closure ofR.

7.6 Normal, separable and Galois extensionsAn algebraic extension L/K is called normal if every irreducible polynomial in K[X] that has a root in L completelyfactors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension eldof F such that L/K is normal and which is minimal with this property.An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable,i.e., has no repeated roots in an algebraic closure over K. A Galois extension is a eld extension that is both normaland separable.A consequence of the primitive element theorem states that every nite separable extension has a primitive element(i.e. is simple).Given any eld extensionL/K, we can consider its automorphismgroupAut(L/K), consisting of all eld automorphisms: L L with (x) = x for all x in K. When the extension is Galois this automorphism group is called the Galoisgroup of the extension. Extensions whose Galois group is abelian are called abelian extensions.For a given eld extension L/K, one is often interested in the intermediate elds F (subelds of L that contain K).The signicance of Galois extensions and Galois groups is that they allow a complete description of the intermediateelds: there is a bijection between the intermediate elds and the subgroups of the Galois group, described by thefundamental theorem of Galois theory.

24 CHAPTER 7. FIELD EXTENSION

7.7 GeneralizationsField extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer non-commutative analog are central simple algebras (CSAs) ring extensions over a eld, which are simple algebra (nonon-trivial 2-sided ideals, just as for a eld) and where the center of the ring is exactly the eld. For example, the onlynite eld extension of the real numbers is the complex numbers, while the quaternions are a central simple algebraover the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be furthergeneralized to Azumaya algebras, where the base eld is replaced by a commutative local ring.

7.8 Extension of scalarsMain article: Extension of scalars

Given a eld extension, one can "extend scalars" on associated algebraic objects. For example, given a real vectorspace, one can produce a complex vector space via complexication. In addition to vector spaces, one can performextension of scalars for associative algebras dened over the eld, such as polynomials or group algebras and theassociated group representations. Extension of scalars of polynomials is often used implicitly, by just considering thecoecients as being elements of a larger eld, but may also be considered more formally. Extension of scalars hasnumerous applications, as discussed in extension of scalars: applications.

7.9 See also Field theory Glossary of eld theory Tower of elds Primary extension Regular extension

7.10 Notes[1] Wolfram|Alpha input: sqrt(2)+sqrt(3)". Retrieved 2010-06-14.

7.11 References Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised third

ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4

7.12 External links Hazewinkel, Michiel, ed. (2001), Extension of a eld, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

Chapter 8

Finite eld

In mathematics, a nite eld or Galois eld (so-named in honor of variste Galois) is a eld that contains a nitenumber of elements. As with any eld, a nite eld is a set on which the operations of multiplication, addition,subtraction and division are dened and satisfy certain basic rules. The most common examples of nite elds aregiven by the integers mod n when n is a prime number.The number of elements of a nite eld is called its order. A nite eld of order q exists if and only if the order q isa prime power pk (where p is a prime number and k is a positive integer). All elds of a given order are isomorphic.In a eld of order pk, adding p copies of any element always results in zero; that is, the characteristic of the eld is p.In a nite eld of order q, the polynomial Xq X has all q elements of the nite eld as roots. The non-zero elementsof a nite eld form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powersof a single element called a primitive element of the eld (in general there will be several primitive elements for agiven eld.)A eld has, by denition, a commutative multiplication operation. A more general algebraic structure that satisesall the other axioms of a eld but isn't required to have a commutative multiplication is called a division ring (orsometimes skeweld). A nite division ring is a nite eld by Wedderburns little theorem. This result shows that theniteness condition in the denition of a nite eld can have algebraic consequences.Finite elds are fundamental in a number of areas of mathematics and computer science, including number theory,algebraic geometry, Galois theory, nite geometry, cryptography and coding theory.

Commutative rings integral domains integrally closed domains unique factorization do-mains principal ideal domains Euclidean domains elds nite elds

8.1 Denitions, rst examples, and basic propertiesA nite eld is a nite set on which the four operations multiplication, addition, subtraction and division (excludingby zero) are dened, satisfying the rules of arithmetic known as the eld axioms. The simplest examples of niteelds are the prime elds: for each prime number p, the eld GF(p) (also denoted Z/pZ, Fp , or Fp) of order (that is,size) p is easily constructed as the integers modulo p.The elements of a prime eld may be represented by integers in the range 0, ..., p 1. The sum, the dierence andthe product are computed by taking the remainder by p of the integer result. The multiplicative inverse of an elementmay be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm Modular integers).Let F be a nite eld. For any element x in F and any integer n, let us denote by nx the sum of n copies of x. Theleast positive n such that n1 = 0 must exist and is prime; it is called the characteristic of the eld.If the characteristic of F is p, the operation (k; x) 7! k xmakes F a GF(p)-vector space. It follows that the numberof elements of F is pn.For every prime number p and every positive integer n, there are nite elds of order pn, and all these elds areisomorphic (see Existence and uniqueness below). One may therefore identify all elds of order pn, which aretherefore unambiguously denoted Fpn , Fpn or GF(pn), where the letters GF stand for Galois eld.[1]

25

26 CHAPTER 8. FINITE FIELD

The identity

(x+ y)p = xp + yp

is true (for every x and y) in a eld of characteristic p. (This follows from the fact that all, except the rst and the last,binomial coecients of the expansion of (x+ y)p are multiples of p).For every element x in the prime eld GF(p), one has xp = x (This is an immediate consequence of Fermats littletheorem, and this may be easily proved as follows: the equality is trivially true for x = 0 and x = 1; one obtains theresult for the other elements of GF(p) by applying the above identity to x and 1, where x successively takes the values1, 2, ..., p 1 modulo p.) This implies the equality

Xp X =Y

a2GF(p)(X a)

for polynomials over GF(p). More generally, every element in GF(pn) satises the polynomial equation xpn x = 0.Any nite eld extension of a nite eld is separable and simple. That is, if E is a nite eld and F is a subeld ofE, then E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon,nite elds are perfect.

8.2 Existence and uniquenessLet q = pn be a prime power, and F be the splitting eld of the polynomial

P = Xq Xover the prime eld GF(p). This means that F is a nite eld of lowest order, in which P has q distinct roots (theroots are distinct, as the formal derivative of P is equal to 1). Above identity shows that the sum and the product oftwo roots of P are roots of P, as well as the multiplicative inverse of a root of P. In other word, the roots of P form aeld of order q, which is equal to F by the minimality of the splitting eld.The uniqueness up to isomorphism of splitting elds implies thus that all elds of order q are isomorphic.In summary, we have the following classication theorem rst proved in 1893 by E. H. Moore:[2]

The order of a nite eld is a prime power. For every prime power q there are elds of orderq, and they are all isomorphic. In these elds, every element satises

xq = x;

and the polynomial Xq X factors as

Xq X =Ya2F

(X a):

It follows that GF(pn) contains a subeld isomorphic to GF(pm) if and only if m is a divisor of n; in that case, thissubeld is unique. In fact, the polynomial Xpm X divides Xpn X if and only if m is a divisor of n.

8.3 Explicit construction of nite elds

8.3.1 Non-prime eldsGiven a prime power q = pn with p prime and n > 1, the eld GF(q) may be explicitly constructed in the followingway. One chooses rst an irreducible polynomial P in GF(p)[X] of degree n (such an irreducible polynomial alwaysexists). Then the quotient ring

8.3. EXPLICIT CONSTRUCTION OF FINITE FIELDS 27

GF(q) = GF(p)[X]/(P )

of the polynomial ring GF(p)[X] by the ideal generated by P is a eld of order q.More explicitly, the elements of GF(q) are the polynomials over GF(p) whose degree is strictly less than n. Theaddition and the subtraction are those of polynomials over GF(p). The product of two elements is the remainderof the Euclidean division by P of the product in GF(p)[X]. The multiplicative inverse of a non-zero element maybe computed with the extended Euclidean algorithm; see Extended Euclidean algorithm Simple algebraic eldextensions.Except in the construction of GF(4), there are several possible choices for P, which produce isomorphic results. Tosimplify the Euclidean division, for P one commonly chooses polynomials of the form

Xn + aX + b;

which make the needed Euclidean divisions very ecient. However, for some elds, typically in characteristic 2,irreducible polynomials of the formXn+ aX + bmay not exist. In characteristic 2, if the polynomial Xn + X + 1 isreducible, it is recommended to choose Xn + Xk + 1 with the lowest possible k that makes the polynomial irreducible.If all these trinomials are reducible, one chooses pentanomials Xn + Xa + Xb + Xc + 1, as polynomials of degreegreater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root.[3]

In the next sections, we will show how this general construction method works for small nite elds.

8.3.2 Field with four elementsOver GF(2), there is only one irreducible polynomial of degree 2:

X2 +X + 1

Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and

GF(4) = GF(2)[X]/(X2 +X + 1):

If one denotes a a root of this polynomial in GF(4), the tables of the operations in GF(4) are the following. Thereis no table for subtraction, as, in every eld of characteristic 2, subtraction is identical to addition. In the third table,for the division of x by y, x must be read on the left, and y on the top.

8.3.3 GF(p2) for an odd prime pFor applying above general construction of nite elds in the case of GF(p2), one has to nd an irreducible polynomialof degree 2. For p = 2, this has been done in the preceding section. If p is an odd prime, there are always irreduciblepolynomials of the form X2 r, with r in GF(p).More precisely, the polynomial X2 r is irreducible over GF(p) if and only if r is a quadratic non-residue modulop (this is almost the denition of a quadratic non-residue). There are p12 quadratic non-residues modulo p. Forexample, 2 is a quadratic non-residue for p = 3, 5, 11, 13, ..., and 3 is a quadratic non-residue for p = 5, 7, 17, .... Ifp 3 mod 4, that is p = 3, 7, 11, 19, ..., one may choose 1 p 1 as a quadratic non-residue, which allows us tohave a very simple irreducible polynomial X2 + 1.Having chosen a quadratic non-residue r, let be a symbolic square root of r, that is a symbol which has the property2 = r, in the same way as the complex number i is a symbolic square root of 1. Then, the elements of GF(p2) areall the linear expressions

a+ b;

28 CHAPTER 8. FINITE FIELD

with a and b in GF(p). The operations on GF(p2) are dened as follows (the operations between elements of GF(p)represented by Latin letters are the operations in GF(p)):

(a+ b) = a+ (b)(a+ b) + (c+ d) = (a+ c) + (b+ d)

(a+ b)(c+ d) = (ac+ rbd) + (ad+ bc)

(a+ b)1 = a(a2 rb2)1 + (b)(a2 rb2)1

8.3.4 GF(8) and GF(27)The polynomial

X3 X 1is irreducible over GF(2) and GF(3), that is, it is irreducible modulo 2 and 3 (to show this it suce to show that it hasno root in GF(2) nor in GF(3)). It follows that the elements of GF(8) and GF(27) may be represented by expressions

a+ b+ c2;

where a, b, c are elements of GF(2) or GF(3) (respectively), and is a symbol such that

3 = + 1:

The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be dened as follows; in followingformulas, the operations between elements of GF(2) or GF(3), represented by Latin letters are the operations in GF(2)or GF(3), respectively:

(a+ b+ c2) = a+ (b)+ (c)2 (ForGF (8); identity) the is operation this(a+ b+ c2) + (d+ e+ f2) = (a+ d) + (b+ e)+ (c+ f)2

(a+ b+ c2)(d+ e+ f2) = (ad+ bf + ce) + (ae+ bd+ bf + ce+ cf)+ (af + be+ cd+ cf)2

8.3.5 GF(16)The polynomial

X4 +X + 1

is irreducible over GF(2), that is, it is irreducible modulo 2. It follows that the elements of GF(16) may be representedby expressions

a+ b+ c2 + d3;

where a, b, c, d are either 0 or 1 (elements of GF(2)), and is a symbol such that

4 = + 1:

As the characteristic of GF(2) is 2, each element is its additive inverse in GF(16). The addition and multiplicationon GF(16) may be dened as follows; in following formulas, the operations between elements of GF(2), representedby Latin letters are the operations in GF(2).

8.4. MULTIPLICATIVE STRUCTURE 29