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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 | ||
A. C. C. see ascending
chain condition Abel's Theorem 308
abelian field extension 292
abelian group 24
—defined by generators
and relations 343ff divisible— 195ff finitely generated— 76ff free— 7 Iff action of group on set 88ff additive notation for
groups 25, 29, 38, 70
adjoining a root 236
adjoint —associativity 214
classical—matrix 353
—pair of functors 477ff affine variety 409
algebra(s) 226ff, 450ff division— 456ff Fundamental Theorem
of— 266
group— 227
homomorphism of—
228
—ideal 228
left Artinian— 451
tensor product of— 229
algebraic —algebra 453
—closure 259ff —element 233, 453
—field extension 233ff —number 242
—set 409
algebraically —closed field 258
—{independent 31 Iff algorithm division— 11, 158
Euclidean— 141
alternating —group 49
—multilinear function 349
annihilator 206, 388,417
anti-isomorphism 330
antisymmetric relation 13
Aitin, E. vi, 251,252, 288
Artinian left—algebras 451
—module 372fT
—rings 372ff, 419ff, 435ff ascending —central series 100
—chain condition 83, 372ff associate 135
associated prime ideal 380,
387
associative —binary operation 24
generalized—law 28
AutkF 243
automorphism 30, 119
extendible— 251, 260
inner—91,459
A^-automorphism 243
—of a group 30, 90
axiom —of choice 13
—of class formation 2
—of extensionality 2
—of pair formation 6
power— 3
Baer lower radical 444ff base
transcendence— 313ff basis 71, 181
dual— 204
Hilbert—Theorem 391
matrix relative to a—
329ff standard— 336
transcendence— 313ff
bijective function 5
bilinear map 211, 349
canonical— 211
bimodule 202
binary operation 24
closed under a— 31
binomial —coefficient 118
—theorem 118
Boolean ring 120
bound (greatest) lower— 14
least upper— 14
upper— 13
cancellation 24, 116
canonical —bilinear map 211
—epimorphism 43, 125,
172
—forms 337
—injection 60, 130, 173
Jordan—form 361
—middle linear map 209
primary rational—form 361
—projection 59,130,173
rational—form 360
Cardan's formulas 310
cardinal number(s) 16ff exponentiation of— 22
—inequalities 17
product of— 16
sum of— 16
trichotomy law for— 18
cardinality 16
Cartesian product 6, 7ff category 52ff, 464ff concrete 55
dual— 466
opposite— 466
product— 467
Cauchy's Theorem 93
Cayley's Theorem 90
