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| Preface | ix | |
| Acknowledgments | xiii | |
| Suggestions on the Use of This Book | xv | |
| Introduction: Prerequisites and Preliminaries | 1 | |
| 1. | Logic | 1 |
| 2. | Sets and Classes | 1 |
| 3. | Functions | 3 |
| 4. | Relations and Partitions | 6 |
| 5. | Products | 7 |
| 6. | The Integers | 9 |
| 7. | The Axiom of Choice, Order and Zorn's Lemma | 12 |
| 8. | Cardinal Numbers | 15 |
| Chapter I: Groups | 23 | |
| 1. | Semigroups, Monoids and Groups | 24 |
| 2. | Homomorphisms and Subgroups | 30 |
| 3. | Cyclic Groups | 35 |
| 4. | Cosets and Counting | 37 |
| 5. | Normality, Quotient Groups, and Homomorphisms | 41 |
| 6. | Symmetric, Alternating, and Dihedral Groups | 46 |
| 7. | Categories: Products, Coproducts, and Free Objects | 52 |
| 8. | Direct Products and Direct Sums | 59 |
| 9. | Free Groups, Free Products, Generators & Relations | 64 |
| Chapter II: The Structure of Groups | 70 | |
| 1. | Free Abelian Groups | 70 |
| 2. | Finitely Generated Abelian Groups | 76 |
| 3. | The Krull-Schmidt Theorem | 83 |
| 4. | The Action of a Group on a Set | 88 |
| 5. | The Sylow Theorems | 92 |
| 6. | Classification of Finite Groups | 96 |
| 7. | Nilpotent and Solvable Groups | 100 |
| 8. | Normal and Subnormal Series | 107 |
| Chapter III: Rings | 114 | |
| 1. | Rings and Homomorphisms | 115 |
| 2. | Ideals | 122 |
| 3. | Factorization in Commutative Rings | 135 |
| 4. | Rings of Quotients and Localization | 142 |
| 5. | Rings of Polynomials and Formal Power Series | 149 |
| 6. | Factorization in Polynomial Rings | 157 |
| Chapter IV: Modules | 168 | |
| 1. | Modules, Homomorphisms and Exact Sequences | 169 |
| 2. | Free Modules and Vector Spaces | 180 |
| 3. | Projective and Iiyective Modules | 190 |
| 4. | Horn and Duality | 199 |
| 5. | Tensor Products | 207 |
| 6. | Modules over a Principal Ideal Domain | 218 |
| 7. | Algebras | 226 |
| Chapter V: Fields and Galois Theory | 230 | |
| 1. | Field Extensions | 231 |
| Appendix: Ruler and Compass Constructions | 238 | |
| 2. | The Fundamental Theorem | 243 |
| Appendix: Symmetric Rational Functions | 252 | |
| 3. | Splitting Fields, Algebraic Closure and Normality | 257 |
| Appendix: The Fundamental Theorem of Algebra | 265 | |
| 4. | The Galois Group of a Polynomial | 269 |
| 5. | Finite Fields | 278 |
| 6. | Separability | 282 |
| 7. | Cyclic Extensions | 289 |
| 8. | Cyclotomic Extensions | 297 |
| 9. | Radical Extensions | 302 |
| Appendix: The General Equation of Degree n | 307 | |
| Chapter VI: The Structure of Fields | 311 | |
| 1. | Transcendence Bases | 311 |
| 2. | Linear Disjointness and Separability | 318 |
| Chapter VII: Linear Algebra | 327 | |
| 1. | Matrices and Maps | 328 |
| 2. | Rank and Equivalence | 335 |
| Appendix : Abelian Groups Defined by | ||
| Generators and Relations | 343 | |
| 3. | Determinants | 348 |
| 4. | Decomposition of a Single Linear Transformation and Similarity | 355 |
| 5. The Characteristic Polynomial, Eigenvectors and Eigenvalues--------366 | ||
| Chapter VIII: Commutative Rings and Modules | 371 | |
| 1. | Chain Conditions | 372 |
| 2. | Prime and Primary Ideals | 377 |
| 3. | Primary Decomposition | 383 |
| 4. | Noetherian Rings and Modules | 387 |
| 5. | Ring Extensions | 394 |
| 6. | Dedekind Domains | 400 |
| 7. | The Hilbert Nullstellensatz | 409 |
| Chapter IX: The Structure of Rings | 414 | |
| 1. | Simple and Primitive Rings | 415 |
| 2. | The Jacobson Radical | 424 |
| 3. | Semisimple Rings | 434 |
| 4. | The Prime Radical; Prime and Semiprime Rings | 444 |
| 5. | Algebras | 450 |
| 6. | Division Algebras | 456 |
| Chapter X: Categories | 464 | |
| 1. | Functors and Natural Transformations | 465 |
| 2. | Adjoint Functors | 476 |
| 3. | Morphisms | 480 |
| List of Symbols | 485 | |
| Bibliography | 489 | |
| Index | 493 | |
| xxiii | ||
