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Part One The Basic Objects of Algebra 3

Chapter Ⅰ Groups 3

1.Monoids 3

2.Groups 7

3.Normal subgroups 13

4.Cyclic groups 23

5.Operations of a group on a set 25

6.Sylow subgroups 33

7.Direct sums and free abelian groups 36

8.Finitely generated abelian groups 42

9.The dual group 46

10.Inverse limit and completion 49

11.Categories and functors 53

12.Free groups 66

Chapter Ⅱ Rings 83

1.Rings and homomorphisms 83

2.Commutative rings 92

3.Polynomials and group rings 97

4.Localization 107

5.Principal and factorial rings 111

Chapter Ⅲ Modules 117

1.Basic definitions 117

2.The group of homomorphisms 122

3.Direct products and sums of modules 127

4.Free modules 135

5.Vector spaces 139

6.The dual space and dual module 142

7.Modules over principal rings 146

8.Euler-Poincare maps 155

9.The snake lemma 157

10.Direct and inverse limits 159

Chapter Ⅳ Polynomials 173

1.Basic properties for polynomials in one variable 173

2.Polynomials over a factorial ring 180

3.Criteria for irreducibility 183

4.Hilbert’s theorem 186

5.Partial fractions 187

6.Symmetric polynomials 190

7.Mason-Stothers theorem and the abc conjecture 194

8.The resultant 199

9.Power series 205

Part Two Algebraic Equations 223

Chapter Ⅴ Algebraic Extensions 223

1.Finite and algebraic extensions 225

2.Algebraic closure 229

3.Splitting fields and normal extensions 236

4.Separable extensions 239

5.Finite fields 244

6.Inseparable extensions 247

Chapter Ⅵ Galois Theory 261

1.Galois extensions 261

2.Examples and applications 269

3.Roots of unity 276

4.Linear independence of characters 282

5.The norm and trace 284

6.Cyclic extensions 288

7.Solvable and radical extensions 291

8.Abelian Kummer theory 293

9.The equation Xn-a＝0 297

10.Galois cohomology 302

11.Non-abelian Kummer extensions 304

12.Algebraic independence of homomorphisms 308

13.The normal basis theorem 312

14.Infinite Galois extensions 313

15.The modular connection 315

Chapter Ⅶ Extensions of Rings 333

1.Integral ring extensions 333

2.Integral Galois extensions 340

3.Extension of homomorphisms 346

Chapter Ⅷ Transcendental Extensions 355

1.Transcendence bases 355

2.Noether normalization theorem 357

3.Linearly disjoint extensions 360

4.Separable and regular extensions 363

5.Derivations 368

Chapter Ⅸ Algebralc Spaces 377

1.Hilbert’s Nullstellensatz 378

2.Algebraic sets，spaces and varieties 381

3.Projections and elimination 388

4.Resultant systems 401

5.Spec of a ring 405

Chapter Ⅹ Noetherlan Rlngs and Modules 413

1.Basic criteria 413

2.Associated primes 416

3.Primary decomposition 421

4.Nakayama’s lemma 424

5.Filtered and graded modules 426

6.The Hilbert polynomial 431

7.Indecomposable modules 439

Chapter Ⅺ Real Fields 449

1.Ordered fields 449

2.Real fields 451

3.Real zeros and homomorphisms 457

Chapter Ⅻ Absolute Values 465

1.Definitions，dependence，and independence 465

2.Completions 468

3.Finite extensions 476

4.Valuations 480

5.Completions and valuations 486

6.Discrete valuations 487

7.Zeros of polynomials in complete fields 491

Part Three Linear Algebra and Representations 503

Chapter ⅩⅢ Matrices and Linear Maps 503

1.Matrices 503

2.The rank of a matrix 506

3.Matrices and linear maps 507

4.Determinants 511

5.Duality 522

6.Matrices and bilinear forms 527

7.Sesquilinear duality 531

8.The simplicity of SL2（F）/±1 536

9.The group SLn（F），n≥3 540

Chapter ⅩⅣ Representation of One Endomorphism 553

1.Representations 553

2.Decomposition over one endomorphism 556

3.The characteristic polynomial 561

Chapter ⅩⅤ Structure of Bllinear Forms 571

1.Preliminaries，orthogonal sums 571

2.Quadratic maps 574

3.Symmetric forms，orthogonal bases 575

4.Symmetric forms over ordered fields 577

5.Hermitian forms 579

6.The spectral theorem（hermitian case） 581

7.The spectral theorem（symmetric case） 584

8.Alternating forms 586

9.The Pfaffian 588

10.Witt’s theorem 589

11.The Witt group 594

Chapter ⅩⅥ The Tensor Product 601

1.Tensor product 601

2.Basic properties 607

3.Flat modules 612

4.Extension of the base 623

5.Some functorial isomorphisms 625

6.Tensor product of algebras 629

7.The tensor algebra of a module 632

8.Symmetric products 635

Chapter ⅩⅦ Semisimpllclty 641

1.Matrices and linear maps over non-commutative rings 641

2.Conditions defining semisimplicity 645

3.The density theorem 646

4.Semisimple rings 651

5.Simple rings 654

6.The Jacobson radical，base change，and tensor products 657

7.Balanced modules 660

Chapter ⅩⅧ Representations of Finite Groups 663

1.Representations and semisimplicity 663

2.Characters 667

3.1-dimensional representations 671

4.The space of class functions 673

5.Orthogonality relations 677

6.Induced characters 686

7.Induced representations 688

8.Positive decomposition of the regular character 699

9.Supersolvable groups 702

10.Brauer’s theorem 704

11.Field of definition of a representation 710

12.Example：GL2 over a finite field 712

Chapter ⅩⅨ The Alternating Product 731

1.Definition and basic properties 731

2.Fitting ideals 738

3.Universal derivations and the de Rham complex 746

4.The Clifford algebra 749

Part Four Homological Algebra 761

Chapter ⅩⅩ General Homology Theory 761

1.Complexes 761

2.Homology sequence 767

3.Euler characteristic and the Grothendieck group 769

4.Injective modules 782

5.Homotopies of morphisms of complexes 787

6.Derived functors 790

7.Delta-functors 799

8.Bifunctors 806

9.Spectral sequences 814

Chapter ⅩⅪ Finite Free Resolutions 835

1.Special complexes 835

2.Finite free resolutions 839

3.Unimodular polynomial vectors 846

4.The Koszul complex 850

Appendix 1 The Transcendence of e and π 867

Appendix 2 Some Set Theory 875

Bibliography 895

Index 903

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