Algebra. Volume 2. Second Edition

Contents......Page 3
Conventions on terminology......Page 10
Table of interdependence of chapters......Page 11
1.1 Finite, countable and uncountable sets......Page 12
1.2 Zorn\'s lemma and well-ordered sets......Page 19
1.3 Categories......Page 27
1.4 Graphs......Page 32
Further exercises......Page 38
2.1 Definitions; modular and distributive lattices......Page 41
2.2 Chain conditions......Page 49
2.3 Boolean algebras......Page 56
2.4 Möbius functions......Page 65
Further exercises......Page 69
3.1 Fields and their extensions......Page 73
3.2 Splitting fields......Page 80
3.3 The algebraic closure of a field......Page 85
3.4 Separability......Page 88
3.5 Automorphisms of field extensions......Page 91
3.6 The fundamental theorem of Galois theory......Page 96
3.7 Roots of unity......Page 102
3.8 Finite fields......Page 108
3.9 Primitive elements; norm and trace......Page 113
3.10 Galois theory of equations......Page 118
3.11 The solution of equations by radicals......Page 124
Further exercises......Page 132
4.1 The category of modules over a ring......Page 135
4.2 Semisimple modules......Page 141
4.3 Matrix rings......Page 146
4.4 Free modules......Page 151
4.5 Projective and injective modules......Page 157
4.6 Duality of finite abelian groups......Page 163
4.7 The tensor product of modules......Page 166
Further exercises......Page 174
5.1 Algebras: definition and examples......Page 176
5.2 Direct products of rings......Page 181
5.3 The Wedderburn structure theorems......Page 185
5.4 The radical......Page 189
5.5 The tensor product of algebras......Page 194
5.6 The regular representation; norm and trace......Page 198
5.7 Composites of fields......Page 202
Further exercises......Page 206
6.1 Inner product spaces......Page 208
6.2 Orthogonal sums and diagonalization......Page 211
6.3 The orthogonal group of a space......Page 215
6.4 Witt\'s cancellation theorem and the Witt group of a field......Page 219
6.5 Ordered fields......Page 223
6.6 The field of real numbers......Page 226
Further exercises......Page 231
7.1 Basic definitions......Page 232
7.2 The averaging lemma and Maschke\'s theorem......Page 237
7.3 Orthogonality and completeness......Page 240
7.4 Characters......Page 244
7.5 Complex representations......Page 252
7.6 Representations of the symmetric group......Page 258
7.7 Induced representations......Page 264
7.8 Applications: the theorems of Burnside and Frobenius......Page 269
Further exercises......Page 273
8.1 Divisibility and valuations......Page 275
8.2 Absolute values......Page 280
8.3 The p-adic numbers......Page 291
8.4 Integral elements......Page 300
8.5 Extension of valuations......Page 305
Further exercises......Page 313
9.1 Operations on ideals......Page 316
9.2 Prime ideals and factorization......Page 318
9.3 Localization......Page 321
9.4 Noetherian rings......Page 328
9.5 Dedekind domains......Page 330
9.6 Modules over Dedekind domains......Page 340
9.7 Algebraic equations......Page 345
9.8 The primary decomposition......Page 349
9.9 Dimension......Page 356
9.10 The Hilbert Nullstellensatz......Page 361
Further exercises......Page 364
10.1 The transmission of information......Page 367
10.2 Block codes......Page 369
10.3 Linear codes......Page 372
10.4 Cyclic codes......Page 380
10.5 Other codes......Page 385
Further exercises......Page 389
11.1 Monoids and monoid actions......Page 390
11.2 Languages and grammars......Page 394
11.3 Automata......Page 398
11.4 Variable-length codes......Page 406
11.5 Free algebras and formal power series rings......Page 415
Further exercises......Page 424
Bibliography......Page 426
List of notations......Page 429
Index......Page 432