algebra comp content

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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