Part One The Basic Objects of Algebra
Chapter I Groups
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
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
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-Poincar6 maps 155
9. The snake lemma 157
~ 10. Direct and inverse limits 159
Chapter IV Polynomials
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
Chapter V Algebraic Extensions
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
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
1. Integral ring extensions 333
2. Integral Galois extensions 340
3. Extension of homomorphisms 346
Chapter VIII Transcendental Extensions
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 Algebraic Spaces
1. Hilbert's Nullstellensatz 377
2. Algebraic sets, spaces and varieties 381
3. Projections and elimination 388
4. Resultant systems 401
5. Spec of a ring 405
Chapter X Noetherian Rings and Modules
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
1. Ordered fields 449
2. Real fields 451
3. Real zeros and homomorphisms 457
Chapter XII Absolute Values
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
Chapter XIII Matrices and Linear Maps
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
1. Representations 553
2. Decomposition over one endomorphism 556
3. The characteristic polynomial 561
Chapter XV Structure of Bilinear Forms
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
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 Semisimplicity
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
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
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
Chapter XX General Homology Theory
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
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
Appendix 2 Some Set Theory
Bibliography
Index
