Abstract Algebra (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en"> Celine Carstensen, Volkswohlbund Insurance, Dortmund, Germany; Benjamin Fine, Fairfield University, Connecticut, USA; Gerhard Rosenberger, Universität Hamburg, Germany.

Preface 6
Contents 8
1 Groups, Rings and Fields 14
1.1 Abstract Algebra 14
1.2 Rings 15
1.3 Integral Domains and Fields 17
1.4 Subrings and Ideals 19
1.5 Factor Rings and Ring Homomorphisms 22
1.6 Fields of Fractions 26
1.7 Characteristic and Prime Rings 27
1.8 Groups 30
1.9 Exercises 32
2 Maximal and Prime Ideals 34
2.1 Maximal and Prime Ideals 34
2.2 Prime Ideals and Integral Domains 35
2.3 Maximal Ideals and Fields 37
2.4 The Existence of Maximal Ideals 38
2.5 Principal Ideals and Principal Ideal Domains 40
2.6 Exercises 41
3 Prime Elements and Unique Factorization Domains 42
3.1 The Fundamental Theorem of Arithmetic 42
3.2 Prime Elements, Units and Irreducibles 48
3.3 Unique Factorization Domains 51
3.4 Principal Ideal Domains and Unique Factorization 54
3.5 Euclidean Domains 58
3.6 Overview of Integral Domains 64
3.7 Exercises 64
4 Polynomials and Polynomial Rings 66
4.1 Polynomials and Polynomial Rings 66
4.2 Polynomial Rings over Fields 68
4.3 Polynomial Rings over Integral Domains 70
4.4 Polynomial Rings over Unique Factorization Domains 71
4.5 Exercises 78
5 Field Extensions 79
5.1 Extension Fields and Finite Extensions 79
5.2 Finite and Algebraic Extensions 82
5.3 Minimal Polynomials and Simple Extensions 83
5.4 Algebraic Closures 87
5.5 Algebraic and Transcendental Numbers 88
5.6 Exercises 91
6 Field Extensions and Compass and Straightedge Constructions 93
6.1 Geometric Constructions 93
6.2 Constructible Numbers and Field Extensions 93
6.3 Four Classical Construction Problems 96
6.3.1 Squaring the Circle 96
6.3.2 The Doubling of the Cube 96
6.3.3 The Trisection of an Angle 96
6.3.4 Construction of a Regular n-Gon 97
6.4 Exercises 102
7 Kronecker’s Theorem and Algebraic Closures 104
7.1 Kronecker’s Theorem 104
7.2 Algebraic Closures and Algebraically Closed Fields 107
7.3 The Fundamental Theorem of Algebra 113
7.3.1 Splitting Fields 113
7.3.2 Permutations and Symmetric Polynomials 114
7.4 The Fundamental Theorem of Algebra 118
7.5 The Fundamental Theorem of Symmetric Polynomials 122
7.6 Exercises 124
8 Splitting Fields and Normal Extensions 126
8.1 Splitting Fields 126
8.2 Normal Extensions 128
8.3 Exercises 131
9 Groups, Subgroups and Examples 132
9.1 Groups, Subgroups and Isomorphisms 132
9.2 Examples of Groups 134
9.3 Permutation Groups 138
9.4 Cosets and Lagrange’s Theorem 141
9.5 Generators and Cyclic Groups 146
9.6 Exercises 152
10 Normal Subgroups, Factor Groups and Direct Products 154
10.1 Normal Subgroups and Factor Groups 154
10.2 The Group Isomorphism Theorems 159
10.3 Direct Products of Groups 162
10.4 Finite Abelian Groups 164
10.5 Some Properties of Finite Groups 169
10.6 Exercises 173
11 Symmetric and Alternating Groups 174
11.1 Symmetric Groups and Cycle Decomposition 174
11.2 Parity and the Alternating Groups 177
11.3 Conjugation in Sn 180
11.4 The Simplicity of An 181
11.5 Exercises 183
12 Solvable Groups 184
12.1 Solvability and Solvable Groups 184
12.2 Solvable Groups 185
12.3 The Derived Series 188
12.4 Composition Series and the Jordan–Hölder Theorem 190
12.5 Exercises 192
13 Groups Actions and the Sylow Theorems 193
13.1 Group Actions 193
13.2 Conjugacy Classes and the Class Equation 194
13.3 The Sylow Theorems 196
13.4 Some Applications of the Sylow Theorems 200
13.5 Exercises 204
14 Free Groups and Group Presentations 205
14.1 Group Presentations and Combinatorial Group Theory 205
14.2 Free Groups 206
14.3 Group Presentations 211
14.3.1 The Modular Group 213
14.4 Presentations of Subgroups 220
14.5 Geometric Interpretation 222
14.6 Presentations of Factor Groups 225
14.7 Group Presentations and Decision Problems 226
14.8 Group Amalgams: Free Products and Direct Products 227
14.9 Exercises 229
15 Finite Galois Extensions 230
15.1 Galois Theory and the Solvability of Polynomial Equations 230
15.2 Automorphism Groups of Field Extensions 231
15.3 Finite Galois Extensions 233
15.4 The Fundamental Theorem of Galois Theory 234
15.5 Exercises 244
16 Separable Field Extensions 246
16.1 Separability of Fields and Polynomials 246
16.2 Perfect Fields 247
16.3 Finite Fields 249
16.4 Separable Extensions 251
16.5 Separability and Galois Extensions 254
16.6 The Primitive Element Theorem 258
16.7 Exercises 260
17 Applications of Galois Theory 261
17.1 Applications of Galois Theory 261
17.2 Field Extensions by Radicals 261
17.3 Cyclotomic Extensions 265
17.4 Solvability and Galois Extensions 266
17.5 The Insolvability of the Quintic 267
17.6 Constructibility of Regular n-Gons 272
17.7 The Fundamental Theorem of Algebra 274
17.8 Exercises 276
18 The Theory of Modules 278
18.1 Modules Over Rings 278
18.2 Annihilators and Torsion 283
18.3 Direct Products and Direct Sums of Modules 284
18.4 Free Modules 286
18.5 Modules over Principal Ideal Domains 289
18.6 The Fundamental Theorem for Finitely Generated Modules 292
18.7 Exercises 296
19 Finitely Generated Abelian Groups 298
19.1 Finite Abelian Groups 298
19.2 The Fundamental Theorem: p-Primary Components 299
19.3 The Fundamental Theorem: Elementary Divisors 301
19.4 Exercises 307
20 Integral and Transcendental Extensions 308
20.1 The Ring of Algebraic Integers. 308
20.2 Integral ring extensions. 311
20.3 Transcendental field extensions. 315
20.4 The transcendence of e and p 320
20.5 Exercises 323
21 The Hilbert Basis Theorem and the Nullstellensatz 325
21.1 Algebraic Geometry 325
21.2 Algebraic Varieties and Radicals 325
21.3 The Hilbert Basis Theorem 327
21.4 The Hilbert Nullstellensatz 328
21.5 Applications and Consequences of Hilbert’s Theorems 330
21.6 Dimensions 333
21.7 Exercises 338
22 Algebraic Cryptography 339
22.1 Basic Cryptography 339
22.2 Encryption and Number Theory 344
22.3 Public Key Cryptography 348
22.3.1 The Diffie-Hellman Protocol 349
22.3.2 The RSA Algorithm 350
22.3.3 The El-Gamal Protocol. 352
22.3.4 Elliptic Curves and Elliptic Curve Methods 354
22.4 Noncommutative Group based Cryptography 355
22.4.1 Free Group Cryptosystems 358
22.5 Ko-Lee and Anshel-Anshel-Goldfeld Methods 362
22.5.1 The Ko-Lee Protocol 363
22.5.2 The Anshel-Anshel-Goldfeld Protocol 363
22.6 Platform Groups and Braid Group Cryptography 364
22.7 Exercises 369
Bibliography 372
Index 376