Topics in algebra

Title page......Page 1
Copyright page......Page 2
Preface to the 2nd edition......Page 3
Preface to the 1st edition......Page 7
Contents......Page 9
1 Preliminary Notions......Page 13
1.1 Set Theory......Page 14
1.2 Mappings......Page 22
1.3 The Integers......Page 30
2 Group Theory......Page 38
2.1 Definition of a Group......Page 39
2.2 Some Examples of Groups......Page 41
2.3 Some Preliminary Lemmas......Page 45
2.4 Subgroups......Page 49
2.5 A Counting Principle......Page 56
2.6 Normal Subgroups and Quotient Groups......Page 61
2.7 Homomorphisms......Page 66
2.8 Automorphisms......Page 78
2.9 Cayley\'s Theorem......Page 83
2.10 Permutation Groups......Page 87
2.11 Another Counting Principle......Page 94
2.12 Sylow\'s Theorem......Page 103
2.13 Direct Products......Page 115
2.14 Finite Abelian Groups......Page 121
3.1 Definition and Examples of Rings......Page 132
3.2 Some Special Classes of Rings......Page 137
3.3 Homomorphisms......Page 143
3.4 Ideals and Quotient Rings......Page 145
3.5 More Ideals and Quotient Rings......Page 149
3.6 The Field of Quotients of an Integral Domain......Page 152
3.7 Euclidean Rings......Page 155
3.8 A Particular Euclidean Ring......Page 161
3.9 Polynomial Rings......Page 165
3.10 Polynomials over the Rational Field......Page 171
3.11 Polynomial Rings over Commutative Rings......Page 173
4 Vector Spaces and Modules......Page 182
4.1 Elementary Basic Concepts......Page 183
4.2 Linear Independence and Bases......Page 189
4.3 Dual Spaces......Page 196
4.4 Inner Product Spaces......Page 203
4.5 Modules......Page 213
5.1 Extension Fields......Page 219
5.2 The Transcendence of $e$......Page 228
5.3 Roots of Polynomials......Page 231
5.4 Construction with Straightedge and Compass......Page 240
5.5 More About Roots......Page 244
5.6 The Elements of Galois Theory......Page 249
5.7 Solvability by Radicals......Page 262
5.8 Galois Groups over the Rationals......Page 268
6 Linear Transformations......Page 272
6.1 The Algebra of Linear Transformations......Page 273
6.2 Characteristic Roots......Page 282
6.3 Matrices......Page 285
6.4 Canonical Forms: Triangular Form......Page 297
6.5 Canonical Forms: Nilpotent Transformations......Page 304
6.6 Canonical Forms: A Decomposition of $V$: Jordan Form......Page 310
6.7 Canonical Forms: Rational Canonical Form......Page 317
6.8 Trace and Transpose......Page 325
6.9 Determinants......Page 334
6.10 Hermitian, Unitary, and Normal Transformations......Page 348
6.11 Real Quadratic Forms......Page 362
7 Selected Topics......Page 367
7.1 Finite Fields......Page 368
7.2 Wedderburn\'s Theorem on Finite Division Rings......Page 372
7.3 A Theorem of Frobenius......Page 380
7.4 Integral Quaternions and the Four-Square Theorem......Page 383
Index......Page 391
Cover......Page 401