Table of Contents
Preface v
Preface to the second edition vii
1 Groups, rings and fields 1
1.1 Abstract algebra 1
1.2 Rings 2
1.3 Integral domains and fields 3
1.4 Subrings and ideals 6
1.5 Factor rings and ring homomorphisms 9
1.6 Fields of fractions 13
1.7 Characteristic and prime rings 14
1.8 Groups 16
1.9 Exercises 19
2 Maximal and prime ideals 21
2.1 Maximal and prime ideals 21
2.2 Prime ideals and integral domains 22
2.3 Maximal ideals and fields 24
2.4 The existence of maximal ideals 25
2.5 Principal ideals and principal ideal domains 26
2.6 Exercises 28
3 Prime elements and unique factorization domains 29
3.1 The fundamental theorem of arithmetic 29
3.2 Prime elements, units and irreducibles 34
3.3 Unique factorization domains 38
3.4 Principal ideal domains and unique factorization 41
3.5 Euclidean domains 44
3.6 Overview of integral domains 50
3.7 Exercises 50
4 Polynomials and polynomial rings 53
4.1 Polynomials and polynomial rings 53
4.2 Polynomial rings over fields 55
4.3 Polynomial rings over integral domains 57
4.4 Polynomial rings over unique factorization domains 59
4.5 Exercises 65
5 Field extensions 67
5.1 Extension fields and finite extensions 67
5.2 Finite and algebraic extensions 70
5.3 Minimal polynomials and simple extensions 71
5.4 Algebraic closures 74
5.5 Algebraic and transcendental numbers 75
5.6 Exercises 78
6 Field extensions and compass and straightedge constructions 81
6.1 Geometric constructions 81
6.2 Constructible numbers and field extensions 81
6.3 Four classical construction problems 84
6.3.1 Squaring the circle 84
6.3.2 The doubling of the cube 84
6.3.3 The trisection of an angle 84
6.3.4 Construction of a regular n-gon 85
6.4 Exercises 89
7 Kronecker's theorem and algebraic closures 93
7.1 Kronecker's theorem 93
7.2 Algebraic closures and algebraically closed fields 96
7.3 The fundamental theorem of algebra 101
7.3.1 Splitting fields 101
7.3.2 Permutations and symmetric polynomials 102
7.4 The fundamental theorem of algebra 106
7.5 The fundamental theorem of symmetric polynomials 109
7.6 Skew field extensions of C and Frobenius's theorem 112
7.7 Exercises 116
8 Splitting fields and normal extensions 119
8.1 Splitting fields 119
8.2 Normal extensions 121
8.3 Exercises 124
9 Groups, subgroups, and examples 125
9.1 Groups, subgroups, and Isomorphisms 125
9.2 Examples of groups 127
9.3 Permutation groups 130
9.4 Cosets and Lagrange's theorem 133
9.5 Generators and cyclic groups 138
9.6 Exercises 144
10 Normal subgroups, factor groups, and direct products 147
10.1 Normal subgroups and factor groups 147
10.2 The group isomorphism theorems 151
10.3 Direct products of groups 155
10.4 Finite Abelian groups 157
10.5 Some properties of finite groups 161
10.6 Automorphisms of a group 165
10.7 Exercises 167
11 Symmetric and alternating groups 169
11.1 Symmetric groups and cycle decomposition 169
11.2 Parity and the alternating groups 172
11.3 Conjugation in Sn 174
11.4 The simplicity of An 175
11.5 Exercises 178
12 Solvable groups 179
12.1 Solvability and solvable groups 179
12.2 Solvable groups 179
12.3 The derived series 183
12.4 Composition series and the Jordan-Holder theorem 185
12.5 Exercises 186
13 Groups actions and the Sylow theorems 189
13.1 Group actions 189
13.2 Conjugacy classes and the class equation 190
13.3 The Sylow theorems 192
13.4 Some applications of the Sylowtheorems 196
13.5 Exercises 200
14 Free groups and group presentations 201
14.1 Group presentations and combinatorial group theory 201
14.2 Free groups 202
14.3 Group presentations 207
14.3.1 The modular group 209
14.4 Presentations of subgroups 215
14.5 Geometric interpretation 218
14.6 Presentations of factor groups 221
14.7 Group presentations and decision problems 222
14.8 Group amalgams: free products and direct products 223
14.9 Exercises 225
15 Finite Galois extensions 227
15.1 Galois theory and the solvability of polynomial equations 227
15.2 Automorphism groups of field extensions 228
15.3 Finite Galois extensions 230
15.4 The fundamental theorem of Galois theory 231
15.5 Exercises 240
16 Separable field extensions 243
16.1 Separability of fields and polynomials 243
16.2 Perfect fields 244
16.3 Finite fields 246
16.4 Separable extensions 247
16.5 Separability and Galois extensions 250
16.6 The primitive element theorem 254
16.7 Exercises 256
17 Applications of Galois theory 257
17.1 Applications of Galois theory 257
17.2 Field extensions by radicals 257
17.3 Cyclotomic extensions 261
17.4 Solvability and Galois extensions 262
17.5 The insolvability of the quintic polynomial 263
17.6 Constructibility of regular n-gons 269
17.7 The fundamental theorem of algebra 271
17.8 Exercises 273
18 The theory of modules 275
18.1 Modules over rings 275
18.2 Annihilators and torsion 279
18.3 Direct products and direct sums of modules 280
18.4 Free modules 282
18.5 Modules over principal ideal domains 285
18.6 The fundamental theorem for finitely generated modules 288
18.7 Exercises 292
19 Finitely generated Abelian groups 293
19.1 Finite Abelian groups 293
19.2 The fundamental theorem: p-primary components 294
19.3 The fundamental theorem: elementary divisors 295
19.4 Exercises 301
20 Integral and transcendental extensions 303
20.1 The ring of algebraic integers 303
20.2 Integral ring extensions 305
20.3 Transcendental field extensions 310
20.4 The transcendence of e and π 315
20.5 Exercises 318
21 The Hilbert basis theorem and the nullstellensatz 319
21.1 Algebraic geometry 319
21.2 Algebraic varieties and radicals 319
21.3 The Hilbert basis theorem 321
21.4 The Hilbert nullstellensatz 322
21.5 Applications and consequences of Hilbert's theorems 323
21.6 Dimensions 326
21.7 Exercises 330
22 Algebras and group representations 333
22.1 Group representations 333
22.2 Representations and modules 334
22.3 Semisimple algebras and Wedderburn's theorem 342
22.4 Ordinary representations, characters and character theory 351
22.5 Burnside's theorem 358
22.6 Exercises 362
23 Algebraic cryptography 365
23.1 Basic cryptography 365
23.2 Encryption and number theory 370
23.3 Public key cryptography 375
23.3.1 The Diffie-Hellman protocol 376
23.3.2 The RSA algorithm 377
23.3.3 The El-Gamal protocol 379
23.3.4 Elliptic curves and elliptic curve methods 381
23.4 Noncommutative-group-based cryptography 382
23.4.1 Free group cryptosystems 384
23.5 Ko-Lee and Anshel-Anshel-Goldfeld methods 389
23.5.1 The Ko-Lee protocol 389
23.5.2 The Anshel-Anshel-Goldfeld protocol 390
23.6 Platform groups and braid group cryptography 391
23.7 Exercises 395
Bibliography 399
Index 403
