| Preface Etymology Special Notation Chapter I Things Past 1.1. Some Number Theory 1.2. Roots of Unity 1.3. Some Set Theory Chapter 2 Groups I 2.1. Introduction 2.2. Permutations 2.3. Groups 2.4. Lagranges Theorem 2.5. Homomorphisms 2.6. Quotient Groups 2.7. Group Actions Chapter 3 Commutative Rings I 3.1. Introduction 3.2. First Properties 3.3. Polynomials 3.4. Greatest Common Divisors 3.5. Homomorphisms 3.6. Euclidean Rings 3.7. Linear Algebra Vector Spaces Linear Transformations 3.8. Quotient Rings and Finite Fields Chapter 4 Fields 4.1. Insolvability of the Quintic Formulas and Solvability by Radicals Translation into Group Theory 4.2. Fundamental Theorem of Galois Theory Chapter 5 Groups II 5.1. Finite Abelian Groups Direct Sums Basis Theorem Fundamental Theorem 5.2. The Sylow Theorems 5.3. The Jordan-Hilder Theorem 5.4. Projective Unimodular Groups 5.5. Presentations 5.6. The Neilsen-Schreier Theorem Chapter 6 Commutative Rings H 6.1. Prime Ideals and Maximal Ideals 6.2. Unique Factorization Domains 6.3. Noetherian Rings 6.4. Applications of Zoms Lemma 6.5. Varieties 6.6. Gr6bner Bases Generalized Division Algorithm Buchbergers Algorithm Chapter 7 Modules and Categories 7.1. Modules 7.2. Categories 7.3. Functors 7.4. Free Modules, Projectives, and Injectives 7.5. Grothendieck Groups 7.6. Limits Chapter 8 Algebras 8.1. Noncommutative Rings 8.2. Chain Conditions 8.3. Semisimple Rings 8.4. Tensor Products 8.5. Characters 8.6. Theorems of Burnside and of Frobenius Contents Chapter 9 Advanced Linear Algebra 9.1. Modules over PIDs 9.2. Rational Canonical Forms 9.3. Jordan Canonical Forms 9.4. Smith Normal Forms 9.5. Bilinear Forms 9.6. Graded Algebras 9.7. Division Algebras 9.8. Exterior Algebra 9.9. Determinants 9.10. Lie Algebras Chapter 10 Homology 10.1. Introduction 10.2. Semidirect Products 10.3. General Extensions and Cohomology 10.4. Homology Functors 10.5. Derived Functors 10.6. Ext and Tor 10.7. Cohomology of Groups 10.8. Crossed Products 10.9. Introduction to Spectral Sequences Chapter 11 Commutative Rings III 11.1. Local and Global 11.2. Dedekind Rings Integrality Nullstellensatz Redux Algebraic Integers Characterizations of Dedekind Rings Finitely Generated Modules over Dedekind Rings 11.3. Global Dimension 11.4. Regular Local Rings AppendixThe Axiom of Choice and Zorns Lemma Bibliography Index |
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