Joseph J. Rotman
Preface to Second Edition ixSpecial Notation xiiiChapter 1. Groups I 11.1. Classical Formulas 11.2. Permutations 51.3. Groups 161.4. Lagrange’s Theorem 281.5. Homomorphisms 381.6. Quotient Groups 471.7. Group Actions 601.8. Counting 76Chapter 2. Commutative Rings I 812.1. First Properties 812.2. Polynomials 912.3. Homomorphisms 962.4. From Arithmetic to Polynomials 1022.5. Irreducibility 1152.6. Euclidean Rings and Principal Ideal Domains 1232.7. Vector Spaces 1332.8. Linear Transformations and Matrices 1452.9. Quotient Rings and Finite Fields 156Chapter 3. Galois Theory 1733.1. Insolvability of the Quintic 173vvi Contents3.1.1. Classical Formulas and Solvability by Radicals 1813.1.2. Translation into Group Theory 1843.2. Fundamental Theorem of Galois Theory 1923.3. Calculations of Galois Groups 212Chapter 4. Groups II 2234.1. Finite Abelian Groups 2234.1.1. Direct Sums 2234.1.2. Basis Theorem 2304.1.3. Fundamental Theorem 2364.2. Sylow Theorems 2434.3. Solvable Groups 2524.4. Projective Unimodular Groups 2634.5. Free Groups and Presentations 2704.6. Nielsen–Schreier Theorem 285Chapter 5. Commutative Rings II 2955.1. Prime Ideals and Maximal Ideals 2955.2. Unique Factorization Domains 3025.3. Noetherian Rings 3125.4. Zorn’s Lemma and Applications 3165.4.1. Zorn’s Lemma 3175.4.2. Vector Spaces 3215.4.3. Algebraic Closure 3255.4.4. L¨uroth’s Theorem 3315.4.5. Transcendence 3355.4.6. Separability 3425.5. Varieties 3485.5.1. Varieties and Ideals 3495.5.2. Nullstellensatz 3545.5.3. Irreducible Varieties 3585.5.4. Primary Decomposition 3615.6. Algorithms in k[x1, . . . , xn] 3695.6.1. Monomial Orders 3705.6.2. Division Algorithm 3765.7. Gr¨obner Bases 3795.7.1. Buchberger’s Algorithm 381Chapter 6. Rings 3916.1. Modules 3916.2. Categories 4186.3. Functors 4376.4. Free and Projective Modules 450Contents vii6.5. Injective Modules 4606.6. Tensor Products 4696.7. Adjoint Isomorphisms 4886.8. Flat Modules 4936.9. Limits 4986.10. Adjoint Functors 5146.11. Galois Theory for Infinite Extensions 518Chapter 7. Representation Theory 5257.1. Chain Conditions 5257.2. Jacobson Radical 5347.3. Semisimple Rings 5397.4. Wedderburn–Artin Theorems 5507.5. Characters 5637.6. Theorems of Burnside and of Frobenius 5907.7. Division Algebras 6007.8. Abelian Categories 6147.9. Module Categories 626Chapter 8. Advanced Linear Algebra 6358.1. Modules over PIDs 6358.1.1. Divisible Groups 6468.2. Rational Canonical Forms 6558.3. Jordan Canonical Forms 6648.4. Smith Normal Forms 6718.5. Bilinear Forms 6828.5.1. Inner Product Spaces 6828.5.2. Isometries 6948.6. Graded Algebras 7048.6.1. Tensor Algebra 7068.6.2. Exterior Algebra 7158.7. Determinants 7298.8. Lie Algebras 743Chapter 9. Homology 7519.1. Simplicial Homology 7519.2. Semidirect Products 7579.3. General Extensions and Cohomology 7659.3.1. H2(Q,K) and Extensions 7669.3.2. H1(Q,K) and Conjugacy 7749.4. Homology Functors 782viii Contents9.5. Derived Functors 7969.5.1. Left Derived Functors 7979.5.2. Right Derived Covariant Functors 8089.5.3. Right Derived Contravariant Functors 8119.6. Ext 8159.7. Tor 8259.8. Cohomology of Groups 8319.9. Crossed Products 8489.10. Introduction to Spectral Sequences 8549.11. Grothendieck Groups 8589.11.1. The Functor K0 8589.11.2. The Functor G0 862Chapter 10. Commutative Rings III 87310.1. Local and Global 87310.1.1. Subgroups of Q 87310.2. Localization 88110.3. Dedekind Rings 89910.3.1. Integrality 90010.3.2. Nullstellensatz Redux 90810.3.3. Algebraic Integers 91510.3.4. Characterizations of Dedekind Rings 92710.3.5. Finitely Generated Modules over Dedekind Rings 93710.4. Homological Dimensions 94510.5. Hilbert’s Theorem on Syzygies 95610.6. Commutative Noetherian Rings 96110.7. Regular Local Rings 969Bibliography 985Index 991