Algebra Volume 1, 2nd.edition

Contents......Page 7
Table of interdependence of chapters......Page 10
1.1 The need for logic......Page 11
1.2 Sets......Page 18
1.3 Mappings......Page 22
1.4 Equivalence relations......Page 27
1.5 Ordered sets......Page 29
Further exercises......Page 31
2.1 The integers......Page 32
2.2 Divisibility and factorization in Z......Page 36
2.3 Congruences......Page 40
2.4 The rational numbers and some finite fields......Page 47
Further exercises......Page 49
3.1 Monoids......Page 52
3.2 Groups; the axioms......Page 55
3.3 Group actions and coset decompositions......Page 61
3.4 Cyclic groups......Page 66
3.5 Permutation groups......Page 68
3.6 Symmetry......Page 73
Further exercises......Page 77
4 Vector spaces and linear mappings......Page 79
4.1 Vectors and linear dependence......Page 80
4.2 Linear mappings......Page 84
4.3 Bases and dimension......Page 86
4.4 Direct sums and quotient spaces......Page 91
4.5 The space of linear mappings......Page 96
4.6 Change of basis......Page 103
4.7 The rank......Page 106
4.8 Affine spaces......Page 111
4.9 Category and functor......Page 114
Further exercises......Page 120
5.1 Systems of linear equations......Page 122
5.2 Elementary operations......Page 125
5.3 Linear programming......Page 130
5.4 PAQ-reduction and the inversion of matrices......Page 139
5.5 Block multiplication......Page 142
Further exercises......Page 144
6.1 Definition and examples......Page 146
6.2 The field of fractions of an integral domain......Page 150
6.3 The characteristic......Page 155
6.4 Polynomials......Page 157
6.5 Factorization......Page 163
6.6 The zeros of polynomials......Page 171
6.7 The factorization of polynomials......Page 174
6.8 Derivatives......Page 178
6.9 Symmetric and alternating functions......Page 187
Further exercises......Page 192
7.1 Definition and basic properties......Page 196
7.2 Expansion of a determinant......Page 204
7.3 The determinantal rank......Page 209
7.4 The resultant......Page 211
Further exercises......Page 215
8.1 Bilinear forms and pairings......Page 218
8.2 Dual spaces......Page 221
8.3 Inner products; quadratic and hermitian forms......Page 226
8.4 Euclidean and unitary spaces......Page 234
8.5 Orthogonal and unitary matrices......Page 240
8.6 Alternating forms......Page 251
Further exercises......Page 255
9.1 The isomorphism theorems......Page 259
9.2 The Jordan-Holder theorem......Page 265
9.3 Groups with operators......Page 269
9.4 Automorphisms......Page 273
9.5 The derived group; soluble groups and simple groups......Page 276
9.6 Direct products......Page 282
9.7 Abelian groups......Page 289
9.8 The Sylow theorems......Page 297
9.9 Generators and defining relations; free groups......Page 302
Further exercises......Page 308
10.1 Ideals and quotient rings......Page 311
10.2 Modules over a ring......Page 314
10.3 Direct products and direct sums......Page 321
10.4 Free modules......Page 324
10.5 Principal ideal domains......Page 328
10.6 Modules over a principal ideal domain......Page 336
Further exercises......Page 342
11 Normal forms for matrices......Page 344
11.1 Eigenvalues and eigenvectors......Page 345
11.2 The k[x]-module defined by an endomorphism......Page 349
11.3 Cyclic endomorphisms......Page 354
11.4 The Jordan normal form......Page 357
11.5 The Jordan normal form: another method......Page 362
11.6 Normal matrices......Page 366
11.7 Linear algebras......Page 368
Further exercises......Page 370
Solutions to the exercises......Page 372
1 Further reading......Page 410
2 Some frequently used notations......Page 411
Index......Page 412