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| Preface | |
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| Preliminaries | |
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| Basics | |
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| Properties of the Integers | |
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| Z / n Z: The Integers Modulo n | |
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| Group Theory | |
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| Introduction to Groups | |
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| Basic Axioms and Examples | |
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| Dihedral Groups | |
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| Symmetric Groups | |
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| Matrix Groups | |
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| The Quaternion Group | |
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| Homomorphisms and Isomorphisms | |
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| Group Actions | |
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| Subgroups | |
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| Definition and Examples | |
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| Centralizers and Normalizers, Stabilizers and Kernels | |
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| Cyclic Groups and Cyclic Subgroups | |
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| Subgroups Generated by Subsets of a Group | |
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| The Lattice of Subgroups of a Group | |
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| Quotient Groups and Homomorphisms | |
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| Definitions and Examples | |
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| More on Cosets and Lagrange's Theorem | |
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| The Isomorphism Theorems | |
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| Composition Series and the Holder Program | |
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| Transpositions and the Alternating Group | |
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| Group Actions | |
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| Group Actions and Permutation Representations | |
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| Groups Acting on Themselves by Left Multiplication--Cayley's Theorem | |
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| Groups Acting on Themselves by Conjugation--The Class Equation | |
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| Automorphisms | |
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| The Sylow Theorems | |
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| The Simplicity of A[subscript n] | |
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| Direct and Semidirect Products and Abelian Groups | |
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| Direct Products | |
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| The Fundamental Theorem of Finitely Generated Abelian Groups | |
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| Table of Groups of Small Order | |
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| Recognizing Direct Products | |
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| Semidirect Products | |
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| Further Topics in Group Theory | |
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| p-groups, Nilpotent Groups, and Solvable Groups | |
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| Applications in Groups of Medium Order | |
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| A Word on Free Groups | |
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| Ring Theory | |
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| Introduction to Rings | |
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| Basic Definitions and Examples | |
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| Examples: Polynomial Rings, Matrix Rings, and Group Rings | |
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| Ring Homomorphisms an Quotient Rings | |
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| Properties of Ideals | |
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| Rings of Fractions | |
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| The Chinese Remainder Theorem | |
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| Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains | |
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| Euclidean Domains | |
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| Principal Ideal Domains (P.I.D.s) | |
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| Unique Factorization Domains (U.F.D.s) | |
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| Polynomial Rings | |
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| Definitions and Basic Properties | |
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| Polynomial Rings over Fields I | |
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| Polynomial Rings that are Unique Factorization Domains | |
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| Irreducibility Criteria | |
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| Polynomial Rings over Fields II | |
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| Polynomials in Several Variables over a Field and Grobner Bases | |
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| Modules and Vector Spaces | |
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| Introduction to Module Theory | |
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| Basic Definitions and Examples | |
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| Quotient Modules and Module Homomorphisms | |
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| Generation of Modules, Direct Sums, and Free Modules | |
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| Tensor Products of Modules | |
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| Exact Sequences--Projective, Injective, and Flat Modules | |
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| Vector Spaces | |
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| Definitions and Basic Theory | |
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| The Matrix of a Linear Transformation | |
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| Dual Vector Spaces | |
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| Determinants | |
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| Tensor Algebras, Symmetric and Exterior Algebras | |
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| Modules over Principal Ideal Domains | |
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| The Basic Theory | |
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| The Rational Canonical Form | |
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| The Jordan Canonical Form | |
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| Field Theory and Galois Theory | |
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| Field Theory | |
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| Basic Theory of Field Extensions | |
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| Algebraic Extensions | |
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| Classical Straightedge and Compass Constructions | |
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| Splitting Fields and Algebraic Closures | |
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| Separable and Inseparable Extensions | |
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| Cyclotomic Polynomials and Extensions | |
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| Galois Theory | |
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| Basic Definitions | |
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| The Fundamental Theorem of Galois Theory | |
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| Finite Fields | |
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| Composite Extensions and Simple Extensions | |
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| Cyclotomic Extensions and Abelian Extensions over Q | |
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| Galois Groups of Polynomials | |
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| Solvable and Radical Extensions: Insolvability of the Quintic | |
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| Computation of Galois Groups over Q | |
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| Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups | |
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| An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra | |
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| Commutative Rings and Algebraic Geometry | |
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| Noetherian Rings and Affine Algebraic Sets | |
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| Radicals and Affine Varieties | |
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| Integral Extensions and Hilbert's Nullstellensatz | |
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| Localization | |
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| The Prime Spectrum of a Ring | |
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| Artinian Rings, Discrete Valuation Rings, and Dedekind Domains | |
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| Artinian Rings | |
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| Discrete Valuation Rings | |
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| Dedekind Domains | |
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| Introduction to Homological Algebra and Group Cohomology | |
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| Introduction to Homological Algebra--Ext and Tor | |
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| The Cohomology of Groups | |
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| Crossed Homomorphisms and H[superscript 1](G, A) | |
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| Group Extensions, Factor Sets and H[superscript 2](G, A) | |
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| Introduction to the Representation Theory of Finite Groups | |
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| Representation Theory and Character Theory | |
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| Linear Actions and Modules over Group Rings | |
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| Wedderburn's Theorem and Some Consequences | |
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| Character Theory and the Orthogonality Relations | |
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| Examples and Applications of Character Theory | |
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| Characters of Groups of Small Order | |
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| Theorems of Burnside and Hall | |
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| Introduction to the Theory of Induced Characters | |
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| Cartesian Products and Zorn's Lemma | |
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| Category Theory | |
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| Index | |