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PART ONE Noncommutative Algebra 1

CHAPTER 1 Definitions and Examples of Groups 3

CHAPTER 2 Subgroups and Cosets 14

CHAPTER 3 Homomorphisms 30

CHAPTER 4 Group Actions 42

CHAPTER 5 The Sylow Theorems and p-groups 55

CHAPTER 6 Permutation Groups 70

CHAPTER 7 New Groups from Old 83

CHAPTER 8 Solvable and Nilpotent Groups 99

CHAPTER 9 Transfer 115

CHAPTER 10 Operator Groups and Unique Decompositions 129

CHAPTER 11 Module Theory without Rings 142

CHAPTER 12 Rings,Ideals,and Modules 159

CHAPTER 13 Simple Modules and Primitive Rings 177

CHAPTER 14 Artinian Rings and Projective Modules 194

CHAPTER 15 An Introduction to Character Theory 213

PART TWO Commutative Algebra 231

CHAPTER 16 Polynomial Rings,PIDs,and UFDs 233

CHAPTER 17 Field Extensions 254

CHAPTER 18 Galois Theory 274

CHAPTER 19 Separability and Inseparability 293

CHAPTER 20 Cyclotomy and Geometric Constructions 307

CHAPTER 21 Finite Fields 326

CHAPTER 22 Roots,Radicals,and Real Numbers 342

CHAPTER 23 Norms,Traces,and Discriminants 359

CHAPTER 24 Transcendental Extensions 379

CHAPTER 25 The Artin-Schreier Theorem 401

CHAPTER 26 Ideal Theory 418

CHAPTER 27 Noetherian Rings 433

CHAPTER 28 Integrality 453

CHAPTER 29 Dedekind Domains 474

CHAPTER 30 Algebraic Sets and the Nullstellensatz 493

Index 507