algebra comp content
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list of general topics for the algebra comp examTRANSCRIPT
Table of contents
I. Basics
a) Groups
i. Abelian Groups
ii. Subgroups
iii. Cyclic Groups
iv. Generating Sets
v. Cosets and Lagrange’sTheorem
vi. Normality
vii. Commutator Subgroup
viii. Homomorphisms and TheIsomorphism Theorems
ix. The correspondenceTheorem
x. Product Groups
xi. Quotient Groups
xii. Composition Series andJordan-Holder Theorem,p-groups, nilpotentgroups, solvability ingenral, a nice matrixexample in terms of itssolvability
xiii. Dihedral Groups
xiv. Symetric Groups
xv. Matrix Groups
xvi. Quoternion Groups
xvii. Group Actions
xviii. Centralizers, Normalizers,Stabilizers and Kernels
xix. The operation on Cosets
xx. The Counting Formula
xxi. Operations on Subsets
xxii. PermutationRepresentations, Groupsacting on themselvesby Left Multiplication,Cayley’s Theorem
xxiii. Groups Actingon Themselves byConjugation, The ClassEquation, Burnside’sLemma, The number offixed points or Diagonalargument
xxiv. Automorphisms,inner automorphisms,The group of innerautomorphisms as asubgroup of permutationgroup for the actionby conjugation and therelated isomorphism,
xxv. The Sylow Theorems
xxvi. The simplicity of An, forn > 5.
xxvii. Direct and SemidirectProducts , Direct Sums
xxviii. Fundamental Theoremfor Finitely Generatedabelian Groups i.e.Structure Theorem
xxix. Recognizing Type ofDirect Products forabelian groups
xxx. Recognizing DirectProducts in general
xxxi. Semidirect Products
and related examples
b) Rings
i. Examples: PolynomialRings, Matrix Rings,Group Rings
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ii. Ring Homomorphismsand Quotient Rings,Ideals
iii. Adjoining Elements,
iv. Rings of Fractions
v. The Chinese RemainderTheorem
vi. Product Rings
vii. Maximal Ideals,
viii. Factoring Integers,
ix. Unique FactorizationDomains,
x. Gauss’s Lemma
xi. Factoring IntegerPolynomials, GaussPrimes,
xii. Polynomial Rings overFields,
xiii. Polynomial Rings that areUFD s,
xiv. Irreducibility Criteria
xv. Polynomial Rings overFields,
xvi. Ideals in Z[ −5√
]
xvii. Ideal Multiplication
xviii. Factoring Ideals
xix. Prime Ideals and Primeintegers
xx. Division Ring
c) Modules
i. Quotient Modules andModule Homomorphisms
ii. Generation of Modules,Direct Sums, and FreeModules,
iii. Projective, Injective andFlat modules
d) Vector Spaces
i. Subspaces,
ii. Bases and Dimensions,
iii. Computing with Bases
iv. Direct Sums
v. Infinite DimensionalSpaces,
vi. Linear Functions,
vii. Bilinear and QuadraticFunctions
viii. Euclidean Spaces,
ix. Hermitian Spaces,
e) Algebras
f) Matrix Algebras
g) Polynomial Algebra
h) Domain
i) Integral Domain
j) Linear Operators
i. Dimension Formula
ii. Matrix of a linearOperator
iii. Eigenvectors
iv. Characteristic polynomial
v. Triangular and DiagonalForms
vi. Bilinear Forms
vii. Symetric Forms
viii. Hermitian Forms
ix. Orthogonality
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x. Linear operators andBilinear Functions onEuclidean Spaces
xi. Spectral Theorem
xii. Jordan Canonical Form
xiii. Functions of a LinearOperator
xiv. Skew-Symetric Form
k) Euclidean Domain
l) Principal Ideal Domain
i. Modules over PrincipalIdeals Domain andRational Canonical Form
m) Tensor Algebra
i. Tensor Product of VectorSpaces
ii. Grasmann Algebra
n) Linear Groups
i. SO(2) acts on R2.
ii. A very crucial thm forthe other groups to definethe structure of the groupor set that they act onby means of the stabilizerand the group, itself
iii. SO(n) acts on Sn−1
iv. O(n) acts on RPn−1
v. Flags in Rn
vi. Sl(2, R) acts on upperhalf plane
o) Linear Algebra in a Ring
1. Modules and FreeModules
2. Diagonalizing integerMatrices
3. Structure of AbelianGroups
4. Applications to LinearOperators
5. Polynomial Rings inSeveral Variables
p) Group Representations for finiteGroups
i. Basic Definitions
ii. IrreducibleRepresentations
iii. Unitary Representations
iv. Characters
v. One dimensionalCharacters
vi. The RegularRepresentation
vii. Schur’s Lemma
viii. Proof of theOrthogonality Relations
q) Fields
i. The degree of a fieldextension
ii. Adjoining Roots
iii. Finite Fields
iv. Primitive Elements
v. The Fundamental Thm ofAlgebra
vi. Finding the irreduciblepolynomial
vii. Algebraic Extensions
viii. Splitting Fields andAlgebraic Closures
ix. Seperable and inseperableExtensions
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x. Cyclotomic Polynomialsand Extensions
r) Galois Theory
i. The discriminant andbasic definitions
ii. Finite Fields
iii. Fixed Fields
iv. Fundamentl Theorem ofGalois Theory
v. Composite and SimpleExtensions
vi. Cyclotomic Extensionsand Abelian Extensionsover Q
vii. Cubic Equations
viii. Quartic Equations
ix. Roots of Unity
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