Book:David S. Dummit/Abstract Algebra/Third Edition
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David S. Dummit and Richard M. Foote: Abstract Algebra (3rd Edition)
Published $\text {2003}$, Wiley
- ISBN 978-0-471-43334-7
Subject Matter
Contents
- Preface
- Preliminaries
- 0.1 Basics
- 0.2 Properties of the Integers
- 0.3 $\Z/n \Z$: 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 Subset 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 Euclidean 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 Irreducibility 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
- Chaoter 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 Cyclotomic 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 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 $\map {H^1} {G, A}$
- 17.4 Group Extensions, Factor Sets and $\map {H^2} {G, A}$
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
Further Editions
- 1990: David S. Dummit and Richard M. Foote: Abstract Algebra
- 1999: David S. Dummit and Richard M. Foote: Abstract Algebra (2nd ed.)