Preface to Second Edition ...................................... ix
Special Notation ............................................. xiii
Chapter 1 Groups I ............................................. 1
1.1 Classical Formulas ......................................... 1
1.2 Permutations ............................................... 5
1.3 Groups .................................................... 16
1.4 Lagrange's Theorem ........................................ 28
1.5 Homomorphisms ............................................. 38
1.6 Quotient Groups ........................................... 47
1.7 Group Actions ............................................. 60
1.8 Counting .................................................. 76
Chapter 2 Commutative Rings I ................................. 81
2.1 First Properties .......................................... 81
2.2 Polynomials ............................................... 91
2.3 Homomorphisms ............................................. 96
2.4 From Arithmetic to Polynomials ........................... 102
2.5 Irreducibility ........................................... 115
2.6 Euclidean Rings and Principal Ideal Domains .............. 123
2.7 Vector Spaces ............................................ 133
2.8 Linear Transformations and Matrices ...................... 145
2.9 Quotient Rings and Finite Fields ......................... 156
Chapter 3 Galois Theory ...................................... 173
3.1 Insolvability of the Quintic ............................. 173
3.1.1 Classical Formulas and Solvability by Radicals .... 181
3.1.2 Translation into Group Theory ..................... 184
3.2 Fundamental Theorem of Galois Theory ..................... 192
3.3 Calculations of Galois Groups ............................ 212
Chapter 4 Groups II .......................................... 223
4.1 Finite Abelian Groups .................................... 223
4.1.1 Direct Sums ....................................... 223
4.1.2 Basis Theorem ..................................... 230
4.1.3 Fundamental Theorem ............................... 236
4.2 Sylow Theorems ........................................... 243
4.3 Solvable Groups .......................................... 252
4.4 Projective Unimodular Groups ............................. 263
4.5 Free Groups and Presentations ............................ 270
4.6 Nielsen-Schreier Theorem ................................. 285
Chapter 5 Commutative Rings II ............................... 295
5.1 Prime Ideals and Maximal Ideals .......................... 295
5.2 Unique Factorization Domains ............................. 302
5.3 Noetherian Rings ......................................... 312
5.4 Zorn's Lemma and Applications ............................ 316
5.4.1 Zorn's Lemma ...................................... 317
5.4.2 Vector Spaces ..................................... 321
5.4.3 Algebraic Closure ................................. 325
5.4.4 Luroth's Theorem .................................. 331
5.4.5 Transcendence ..................................... 335
5.4.6 Separability ...................................... 342
5.5 Varieties ................................................ 348
5.5.1 Varieties and Ideals .............................. 349
5.5.2 Nullstellensatz ................................... 354
5.5.3 Irreducible Varieties ............................. 358
5.5.4 Primary Decomposition ............................. 361
5.6 Algorithms in k[x1,..., xn] .............................. 369
5.6.1 Monomial Orders ................................... 370
5.6.2 Division Algorithm ................................ 376
5.7 Grobner Bases ............................................ 379
5.7.1 Buchberger's Algorithm ............................ 381
Chapter 6 Rings .............................................. 391
6.1 Modules .................................................. 391
6.2 Categories ............................................... 418
6.3 Functors ................................................. 437
6.4 Free and Projective Modules .............................. 450
6.5 Injective Modules ........................................ 460
6.6 Tensor Products .......................................... 469
6.7 Adjoint Isomorphisms ..................................... 488
6.8 Flat Modules ............................................. 493
6.9 Limits ................................................... 498
6.10 Adjoint Functors ......................................... 514
6.11 Galois Theory for Infinite Extensions .................... 518
Chapter 7 Representation Theory .............................. 525
7.1 Chain Conditions ......................................... 525
7.2 Jacobson Radical ......................................... 534
7.3 Semisimple Rings ......................................... 539
7.4 Wedderburn-Artin Theorems ................................ 550
7.5 Characters ............................................... 563
7.6 Theorems of Burnside and of Frobenius .................... 590
7.7 Division Algebras ........................................ 600
7.8 Abelian Categories ....................................... 614
7.9 Module Categories ........................................ 626
Chapter 8 Advanced Linear Algebra ............................ 635
8.1 Modules over PIDs ........................................ 635
8.1.1 Divisible Groups .................................. 646
8.2 Rational Canonical Forms ................................. 655
8.3 Jordan Canonical Forms ................................... 664
8.4 Smith Normal Forms ....................................... 671
8.5 Bilinear Forms ........................................... 682
8.5.1 Inner Product Spaces .............................. 682
8.5.2 Isometries ........................................ 694
8.6 Graded Algebras .......................................... 704
8.6.1 Tensor Algebra .................................... 706
8.6.2 Exterior Algebra .................................. 715
8.7 Determinants ............................................. 729
8.8 Lie Algebras ............................................. 743
Chapter 9 Homology ........................................... 751
9.1 Simplicial Homology ...................................... 751
9.2 Semidirect Products ...................................... 757
9.3 General Extensions and Cohomology ........................ 765
9.3.1 H2(Q,K) and Extensions ............................ 766
9.3.2 H1(Q,K) and Conjugacy ............................. 774
9.4 Homology Functors ........................................ 782
9.5 Derived Functors ......................................... 796
9.5.1 Left Derived Functors ............................. 797
9.5.2 Right Derived Covariant Functors .................. 808
9.5.3 Right Derived Contravariant Functors .............. 811
9.6 Ext ...................................................... 815
9.7 Tor ...................................................... 825
9.8 Cohomology of Groups ..................................... 831
9.9 Crossed Products ......................................... 848
9.10 Introduction to Spectral Sequences ....................... 854
9.11 Grothendieck Groups ...................................... 858
9.11.1 The Functor K0 .................................... 858
9.11.2 The Functor G0 .................................... 862
Chapter 10 Commutative Rings III .............................. 873
10.1 Local and Global ......................................... 873
10.1.1 Subgroups of  .................................... 873
10.2 Localization ............................................. 881
10.3 Dedekind Rings ........................................... 899
10.3.1 Integrality ....................................... 900
10.3.2 Nullstellensatz Redux ............................. 908
10.3.3 Algebraic Integers ................................ 915
10.3.4 Characterizations of Dedekind Rings ............... 927
10.3.5 Finitely Generated Modules over Dedekind Rings .... 937
10.4 Homological Dimensions ................................... 945
10.5 Hilbert's Theorem on Syzygies ............................ 956
10.6 Commutative Noetherian Rings ............................. 961
10.7 Regular Local Rings ...................................... 969
Bibliography .................................................. 985
Index ......................................................... 991
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