Contents

Preface To The Second Edition Xi

Perface To The First Edition Xiii

Preliminaries 1

1. Basics 1

2. Properties Of The Integers 4

3. Z / n Z The Integers Modulo n 8

Part - Group Theory 13

Chapter 1 Introduction To Groups 16

Chapter 2 Subgroups 47

Chapter 3. Quotient Groups And Homomorphisms 74

Chapter 4. Group Actions 114

Chapter 5 Direct And Semidirect Products And Abelian Groups 154

Chapter 6 Further Topics In Group Theory 190

Part II - Ring Theory 223

Chapter 7 Introduction To Rings 224

Chapter 8. Euclidean Domains Principal Ideal Domains 271

Chapter 9. Polynomial Rings 296

Part III Modules And Vector Spaces 317

Chapter 10 Interoduction To Module Theory 318

Chapter 11. Vector Spaces 388

Chapter 12. Modules Over Principal Ideal Domains 436

Part IV Field Theory And Galois Theory 489

Chapter 13 Field Theory 490

Chapter Galois Theory 538

Part V - An Interoduction To Commitative Rings Algebraic Geometry And Homological Algebra 636

Chapter 15 Commutative Rings And Algebraic Geometry 637

Chapter 16. Artinian Rings Discrete Valuation Rings And Dedekind Domains 716

Chapter 17. Introduction To Homological Algebra And Group Cohology 743

Part VI - Introduction To The Representation Theory Of Finite Groups 805

Chapter 18 Represenation Theory And Character Theory 806

Chapter 19 Examples And Applications Of Charater Theory 846

Appendix Cartesian Products And Zorn's Lemma 871

Appendix II Category Theory 877