# Book:David S. Dummit/Abstract Algebra

## David S. Dummit and Richard M. Foote: Abstract Algebra

Published $\text {1990}$, Prentice Hall

ISBN 978-0130047717

### 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 MultiplicationCayley'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
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
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 - INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS

Chapter 15 Representation Theory and Character Theory
15.1 Linear Actions and Modules over Group Rings
15.2 Projective and Injective Modules
15.3 Statement of Wedderburn's Theorem and Some Consequences
15.4 Character Theory and the Orthogonality Relations
Chapter 16 Examples and Applications of Character Theory
16.1 Characters of Groups of Small Order
16.2 Theorems of Burnside and Hall
16.3 Introduction to the Theory of Induced Characters
Appendix: Cartesian Products and Zorn's Lemma
Index