Book:David S. Dummit/Abstract Algebra/Second Edition

Subject Matter

 * Abstract Algebra

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

'''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 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 and
 * 19.3 Introduction to the Theory of Induced Characters


 * '''Appendix I: Cartesian Products and Zorn's Lemma


 * '''Appendix II: Category Theory


 * Index