About the BookAbout the AuthorTable of Contents

Joseph J. Rotman.

Editorial Board
David Cox (Chair), Steven G. Krantz, Rafe Mazzeo, Martin Scharlemann.

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
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vi Contents
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. L¨uroth’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. Gr¨obner 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
Contents vii
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

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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 Q 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