This book represents a complete course in abstract algebra, providing instructors with flexibility in the selection of topics to be taught in individual classes. All the topics presented are discussed in a direct and detailed manner. Throughout the text, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises. The book contains many examples fully worked out and a variety of problems for practice and challenge. Solutions to the odd-numbered problems are provided at the end of the book. This new edition contains an introduction to lattices, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Lasker–Noether theorem. In addition, there are over 100 new problems and examples, particularly aimed at relating abstract concepts to concrete situations.

Table of Contents

Part I Preliminaries

Chapter I Sets and mappings

Chapter 2 Integers, real numbers, and complex numbers

Chapter 3 Matrices and determinants

Part II Groups

Chapter 4 Groups

Chapter 5 Normal subgroups

Chapter 6 Normal series

Chapter 7 Permutation groups

Chapter 8 Structure theorems of groups

Part III Rings and modules

Chapter 9 Rings

Chapter 10 Ideals and homomorphisms

Chapter 11 Unique factorization domains and euclidean domains

Chapter 12 Rings of fractions

Chapter 13 Integers

Chapter 14 Modules and vector spaces

Chapter 15 Algebraic extensions of fields

Chapter 16 Normal and separable extensions

Chapter 17 Galois theory

Chapter 18 Applications of Galois theory to classical problems

Chapter 19 Noetherian and artinian modules and rings

Chapter 20 Smith normal form over a PID and rank

Chapter 21 Finitely generated modules over a PID

Chapter 22 Tensor products