Basic Abstract Algebra

Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface to the second edition......Page 14
Preface to the first edition......Page 15
Glossary of symbols......Page 19
Part I Preliminaries......Page 22
1.  Sets       ......Page 24
2.  Relations  ......Page 30
3.  Mappings  ......Page 35
4.  Binary operations   ......Page 42
5.  Cardinality of a set......Page 46
1.  Integers  ......Page 51
2.  Rational, real, and complex numbers     ......Page 56
3.  Fields    ......Page 57
1.  Matrices  ......Page 60
2.  Operations on matrices        ......Page 62
3.  Partitions of a matrix        ......Page 67
4.  The determinant function      ......Page 68
5.  Properties of the determinant function  ......Page 70
6.  Expansion of det A  ......Page 74
Part II  Groups......Page 80
I.  Semigroups and groups......Page 82
2.  Homomorphisms        ......Page 90
3.  Subgroups and cosets ......Page 93
4.  Cyclic groups        ......Page 103
5.  Permutation groups   ......Page 105
6.  Generators and relations       ......Page 111
1.  Normal subgroups and quotient groups     ......Page 112
2.  Isomorphism theorems ......Page 118
3.  Automorphisms       ......Page 125
4.  Conjugacy and G-sets......Page 128
1.  Normal series       ......Page 141
2.  Solvable groups     ......Page 145
3.  Nilpotent groups    ......Page 147
1.  Cyclic decomposition......Page 150
2.  Alternating group ......Page 153
3.  Simplicity of     ......Page 156
1.  Direct products     ......Page 159
2.  Finitely generated abelian groups       ......Page 162
3.  Invariants of a finite abelian group    ......Page 164
4.  Sylow theorems      ......Page 167
5.  Groups of orders p2. pq       ......Page 173
Part III   Rings and modules......Page 178
1.  Definition and examples       ......Page 180
2.  Elementary properties of rings......Page 182
3.  Types of rings      ......Page 184
4.  Subrings and characteristic of a ring   ......Page 189
5.  Additional examples of rings  ......Page 197
 1.  Ideals ......Page 200
 2.  Homomorphisms    ......Page 208
 3.  Sum and direct sum of ideals         ......Page 217
 4.  Maximal and prime ideals   ......Page 224
 5.  Nilpotent and nil ideals   ......Page 230
 6.  Zorn's lemma     ......Page 231
 1.  Unique factorization domains         ......Page 233
 2.  Principal ideal domains    ......Page 237
 3.  Euclidean domains......Page 238
 4.  Polynomial rings over UFD  ......Page 240
 1.  Rings of fractions         ......Page 245
 2.  Rings with Ore condition   ......Page 249
 1,  Peano's axioms   ......Page 254
 2.  Integers         ......Page 261
 1.  Definition and examples    ......Page 267
 2.  Submodules and direct sums ......Page 269
 3.  R-homomorphisms and quotient modules ......Page 274
 4.  Completely reducible modules         ......Page 281
 5.  Free modules     ......Page 284
 6.  Representation of linear mappings    ......Page 289
 7.  Rank of a linear mapping   ......Page 294
 1.  Irreducible polynomials and Eisenstein criterion        ......Page 302
 2.  Adjunction of roots        ......Page 306
 3.  Algebraic extensions       ......Page 310
 4.  Algebraically closed fields......Page 316
 1. Splitting fields ......Page 321
 2. Normal extensions......Page 325
 3. Multiple roots   ......Page 328
 4. Finite fields    ......Page 331
 5. Separable extensions       ......Page 337
 1. Automorphism groups and fixed fields ......Page 343
 2. Fundamental theorem of Galois theory ......Page 351
 3. Fundamental theorem of algebra       ......Page 359
 1. Roots of unity and cyclotomic polynomials      ......Page 361
 2. Cyclic extensions......Page 365
 3. Polynomials solvable by radicals     ......Page 369
 4. Symmetric functions        ......Page 376
 5. Ruler and compass constructions      ......Page 379
 1. HomR    ......Page 388
 2. Noetherian and artinian modules      ......Page 389
 3. Wedderburn?rtin theorem   ......Page 402
 4. Uniform modules, primary modules, and Noether?asker theorem     ......Page 408
 1. Preliminaries    ......Page 412
 2. Row module, column module, and rank  ......Page 413
 3. Smith normal form......Page 414
 1. Decomposition theorem      ......Page 422
 2. Uniqueness of the decomposition      ......Page 424
 3. Application to finitely generated abelian groups ......Page 428
 4. Rational canonical form    ......Page 429
 5. Generalized Jordan form over any field         ......Page 437
 1. Categories and functors   ......Page 445
 2. Tensor products ......Page 447
 3. Module structure of tensor product  ......Page 450
 4. Tensor product of homomorphisms     ......Page 452
 5. Tensor product of algebras......Page 455
Solutions to odd-numbered problems       ......Page 457
Selected bibliography......Page 495
Index      ......Page 496
Back Cover......Page 507