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1．Sets 1

1.1 Finite,Countable and Uncountable Sets 1

1.2 Zorn's Lemma and Well-ordered Sets 8

1.3 Graphs 15

2．Groups 25

2.1 Definition and Basic Properties 25

2.2 Permutation Groups 32

2.3 The Isomorphism Theorems 34

2.4 Soluble and Nilpotent Groups 37

2.5 Commutators 42

2.6 The Frattini Subgroup and the Fitting Subgroup 46

3．Lattices and Categories 51

3.1 Definitions;Modular and Distributive Lattices 51

3.2 Chain Conditions 60

3.3 Categories 65

3.4 Boolean Algebras 70

4．Rings and Modules 79

4.1 The Definitions Recalled 79

4.2 The Category of Modules over a Ring 84

4.3 Semisimple Modules 91

4.4 Matrix Rings 96

4.5 Direct Products of Rings 101

4.6 Free Modules 105

4.7 Projective and Injective Modules 110

4.8 The Tensor Product of Modules 117

4.9 Duality of Finite Abelian Groups 125

5．Algebras 131

5.1 Algebras;Definition and Examples 131

5.2 The Wedderburn Structure Theorems 137

5.3 The Radical 141

5.4 The Tensor Product of Algebras 146

5.5 The Regular Representation;Norm and Trace 153

5.6 M？bius Functions 157

6．Multilinear Algebra 165

6.1 Graded Algebras 165

6.2 Free Algebras and Tensor Algebras 168

6.3 The Hilbert Series of a Graded Ring or Module 173

6.4 The Exterior Algebra on a Module 179

7．Field Theory 189

7.1 Fields and their Extensions 189

7.2 Splitting Fields 195

7.3 The Algebraic Closure of a Field 200

7.4 Separability 203

7.5 Automorphisms of Field Extensions 206

7.6 The Fundamental Theorem of Galois Theory 211

7.7 Roots of Unity 217

7.8 Finite Fields 223

7.9 Primitive Elements;Norm and Trace 227

7.10 Galois Theory of Equations 232

7.11 The Solution of Equations by Radicals 238

8．Quadratic Forms and Ordered Fields 249

8.1 Inner Product Spaces 249

8.2 Orthogonal Sums and Diagonalization 252

8.3 The Orthogonal Group of a Space 256

8.4 The Clifford Algebra and the Spinor Norm 259

8.5 Witt's Cancelation Theorem and the Witt Group of a Field 268

8.6 Ordered Fields 272

8.7 The Field of Real Numbers 275

8.8 Formally Real Fields 279

8.9 The Witt Ring of a Field 291

8.10 The Symplectic Group 298

8.11 Quadratic Forms in Characteristic Two 301

9．Valuation Theory 307

9.1 Divisibility and Valuations 307

9.2 Absolute Values 312

9.3 The p-adic Numbers 322

9.4 Integral Elements 331

9.5 Extension of Valuations 336

10．Commutative Rings 347

10.1 Operations on Ideals 347

10.2 Prime Ideals and Factorization 349

10.3 Localization 354

10.4 Noetherian Rings 361

10.5 Dedekind Domains 362

10.6 Modules over Dedekind Domains 371

10.7 Algebraic Equations 376

10.8 The Primary Decomposition 380

10.9 Dimension 386

10.10 The Hilbert Nullstellensatz 391

11．Infinite Field Extensions 397

11.1 Abstract Dependence Relations 397

11.2 Algebraic Dependence 402

11.3 Simple Transcendental Extensions 405

11.4 Separable and p-radical Extensions 409

11.5 Derivations 414

11.6 Linearly Disjoint Extensions 418

11.7 Composites of Fields 427

11.8 Infinite Algebraic Extensions 431

11.9 Galois Descent 437

11.10 Kummer Extensions 441

Bibliography 449

List of Notations 453

Author Index 457

Subject Index 459