Abstract Algebra
Dummit&Foote, Abstract Algebra
Algebra shows structure by equation.
Chapter 0 Preliminaries
0.1 Basics
0.2 Properties of the Integers
0.3 $Z/(n)$ – The Integers Modulo $n$
Chapter 1 Introduction to Groups
1.1 Basic Axioms and Examples
1.2 Dihedral Groups
1.3 Symmetric Groups
1.4 Matrix Groups
1.5 The Quaternion Group
1.6 Homomorphisms and Isomorphisms
1.7 Group Actions
Chapter 2 Subgroups
2.1 Definition and Examples
2.2 Centralizers and Normalizers, Stabilizers and Kernels
2.3 Cyclic Groups and Cyclic Subgroups
2.4 Subgroups Generated by Subsets of a Group
2.5 The Lattice of Subgroups of a Group
Chapter 3: Quotient Groups and Homomorphisms
3.1 Definitions and Examples
3.2 More on Cosets and Lagrange’s Theorem
3.3 The Isomorphism Theorems
3.4 Composition Series and the Hölder Program
3.5 Transpositions and the Alternating Group
Chapter 4 Group Actions
4.1 Group Actions and Permutation Representations
4.2 Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem
4.3 Groups Acting on Themselves by Conjugation – The Class Equation
4.4 Automorphisms
4.5 The Sylow Theorems
4.6 The Simplicity of $A_n$
Chapter 5 Direct and Semidirect Products and Abelian Groups
5.1 Direct Products
5.2 The Fundamental Theorem of Finitely Generated Abelian Groups
5.3 Table of Groups of Small Order
5.4 Recognizing Direct Products
5.5 Semidirect Products
Chapter 6 Further Topics in Group Theory
6.1 p-Groups, Nilpotent Groups, and Solvable Groups
6.2 Applications in Groups of Medium Order
6.3 A Word on Free Groups
Chapter 7 Introduction to Rings
7.1 Basic Definitions and Examples
7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings
7.3 Ring Homomorphisms and Quotient Rings
7.4 Properties of Ideals
7.5 Rings of Fractions
7.6 The Chinese Remainder Theorem
Chapter 8 Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains
8.1 Euclidean Domains
8.2 Principal Ideal Domains
8.3 Unique Factorization Domains
Chapter 9 Polynomial Rings
9.1 Definitions and Basic Properties
9.2 Polynomial Rings over Fields I
9.3 Polynomial Rings that are Unique Factorization Domains
9.4 Irreducibility Criteria
9.5 Polynomial Rings over Fields II
9.6 Polynomials in Several Variables over a Field and Gröbner Bases
Chapter 10 Introduction to Module Theory
10.1 Basic Definitions and Examples
10.2 Quotient Modules and Module Homomorphisms
10.3 Generation of Modules, Direct Sums, and Free Modules
10.4 Tensor Products of Modules
10.5 Exact Sequences – Projective, Injective, and Flat Modules
Chapter 11 Vector Spaces
11.1 Definitions and Basic Theory
11.2 The Matrix of a Linear Transformation
11.3 Dual Vector Spaces
11.4 Determinants
11.5 Tensor Algebras, Symmetric and Exterior Algebras
Chapter 12 Modules over Principal Ideal Domains
12.1 The Basic Theory
12.2 The Rational Canonical Form
12.3 The Jordan Canonical Form
Chapter 13 Field Theory
13.1 Basic Theory of Field Extensions
13.2 Algebraic Extensions
13.3 Classical Straightedge and Compass Constructions
13.4 Splitting Fields and Algebraic Closures
13.5 Separable and Inseparabel Extensions
13.6 Cyclotomic Polynomials and Extensions
Chapter 14 Galois Theory
14.1 Basic Definitions
14.2 The Fundamental Theorem of Galois Theory
14.3 Finite Fields
14.4 Composite Extensions and Simple Extensions
14.5 Cyclotomic Extensions and Abelian Extensions over $\mathbb{Q}$
14.6 Galois Groups of Polynomials
14.7 Solvable and Radical Extensions: Insolvability of the Quintic
14.8 Computation of Galois Groups over $\mathbb{Q}$
14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
Chapter 15 Commutative Rings and Algebraic Geometry
15.1 Noetherian Rings and Affine Algebraic Sets
15.2 Radicals and Affine Varieties
15.3 Integral Extensions and Hilbert’s Nullstellensatz
15.4 Localization
15.5 The Prime Spectrum of a Ring
Chapter 16 Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
16.1 Artinian Rings
16.2 Discrete Valuation Rings
16.3 Dedekind Domains
Chapter 17 Introduction to Homological Algebra and Group Cohomology
17.1 Introduction to Homological Algebra – Ext and Tor
17.2 The Cohomology of Groups
17.3 Crossed Homomorphisms and $H^1(G,A)$
17.4 Group Extensions, Factor Sets, and $H^2(G,A)$
Chapter 18 Representation Theory and Character Theory
18.1 Linear Actions and Modules over Group Rings
18.2 Wedderburn’s Theorem and Some Consequences
18.3 Character Theory and the Orthogonality Relations
Chapter 19 Examples and Applications of Character Theory
19.1 Characters of Groups of Small Order
19.2 Theorems of Burnside and Hall
19.3 Introduction to the Theory of Induced Characters
Reference
Dummit, Foote - Abstract Algebra