Abstract Algebra

Frequently Used Notation

Contents

Preface

Preliminaries

    0.1 Basics

    0.2 Properties of the Integers

    0.3 Z/nZ: The Integers Modulo n

Part I - Group Theory

    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

Part II - Ring Theory

    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 - Euclicdean Domains

        8.2 - Principal Ideal Domains (P.I.D.s)

        8.3 - Unique Factorization Domains (U.F.D.s)

    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 - Irriducibility Criteria

        9.5 - Polynomial Rings over Fields II

        9.6 - Polynomials in Several Variables over a Field and Gröbner Bases

Part III - Modules and Vector Spaces

    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

Part IV - Field Theory and Galois Theory

    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 Inseparable 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 Q

        14.6 - Galois Groups of Polynomials

        14.7 - Solvable and Radical Extensions: Insolvability of the Quintic

        14.8 - Computation of Galois Groups over Q

        14.9 - Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups

Part V - An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra

    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 Hilber\'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)

Part VI - Introduction to the Representation Theory of Finite Groups

    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

Appendix I - Cartesian Products and Zorn\'s Lemma

Appendix II - Category Theory

Index