Galois Theory

Contents (vii)......Page 6
Acknowledgements (xi)......Page 10
Preface to the First Edition (xiii)......Page 11
Preface to the Second Edition (xv)......Page 13
Preface to the Third Edition (xvii)......Page 14
Preface to the Fourth Edition (xxi)......Page 17
Historical Introduction (1)......Page 19
Polynomial Equations (2)......Page 20
The Life of Galois (5)......Page 23
1. Classical Algebra (17)......Page 35
1.2 Subfields and Subrings of the Complex Numbers (18)......Page 36
1.3 Solving Equations (22)......Page 40
1.4 Solution by Radicals (24)......Page 42
Exercises (31)......Page 49
2.1 Polynomials (35)......Page 52
2.2 Fundamental Theorem of Algebra (39)......Page 56
2.3 Implications (42)......Page 59
Exercises (44)......Page 61
3.1 The Euclidean Algorithm (47)......Page 63
3.2 Irreducibility (51)......Page 67
3.3 Gauss's Lemma (54)......Page 70
3.4 Eisenstein's Criterion (55)......Page 71
3.5 Reduction modulo �� (57)......Page 73
3.6 Zeros of Polynomials (58)......Page 74
Exercises (60)......Page 76
4.1 Field Extensions (63)......Page 78
4.2 Rational Expressions (66)......Page 81
4.3 Simple Extensions (67)......Page 82
Exercises (69)......Page 84
5.1 Algebraic and Transcendental Extensions (71)......Page 86
5.2 The Minimal Polynomial (72)......Page 87
5.3 Simple Algebraic Extensions (73)......Page 88
5.4 Classifying Simple Extensions (75)......Page 90
Exercises (77)......Page 92
6.1 Definition of the Degree (79)......Page 94
6.2 The Tower Law (80)......Page 95
Exercises (84)......Page 99
7. Ruler-and-Compass Constructions (87)......Page 102
7.1 Approximate Constructions and More General Instruments (89)......Page 104
7.2 Constructions in ℂ (90)......Page 105
7.3 Specific Constructions (94)......Page 109
7.4 Impossibility Proofs (99)......Page 114
7.5 Construction From a Given Set of Points (101)......Page 116
Exercises (102)......Page 117
8. The Idea Behind Galois Theory (107)......Page 121
8.2 Galois Groups According to Galois (108)......Page 122
8.3 How to Use the Galois Group (110)......Page 124
8.4 The Abstract Setting (111)......Page 125
8.5 Polynomials and Extensions (112)......Page 126
8.6 The Galois Correspondences (114)......Page 128
8.7 Diet Galois (116)......Page 130
8.8 Natural Irrationalities (121)......Page 135
Exercises (125)......Page 139
9.1 Splitting Fields (129)......Page 142
9.2 Normality (132)......Page 145
9.3 Separability (133)......Page 146
Exercises (135)......Page 148
10.1 Linear Independence of Monomorphisms (137)......Page 150
Exercises (142)......Page 155
11.1 ��-Monomorphisms (145)......Page 157
11.2 Normal Closures (146)......Page 158
Exercises (149)......Page 161
12.1 The Fundamental Theorem of Galois Theory (151)......Page 163
Exercises (153)......Page 165
13. A Worked Example (155)......Page 167
Exercises (159)......Page 171
14.1 Soluble Groups (161)......Page 173
14.2 Simple Groups (164)......Page 176
14.3 Cauchy's Theorem (166)......Page 178
Exercises (168)......Page 180
15.1 Radical Extensions (171)......Page 182
15.2 An Insoluble Quintic (176)......Page 187
15.3 Other Methods (178)......Page 189
Exercises (179)......Page 190
16.1 Rings and Fields (181)......Page 192
16.2 General Properties of Rings and Fields (184)......Page 195
16.3 Polynomials Over General Rings (186)......Page 197
16.4 The Characteristic of a Field (187)......Page 198
16.5 Integral Domains (188)......Page 199
Exercises (191)......Page 202
17.1 Minimal Polynomials (193)......Page 204
17.2 Simple Algebraic Extensions (194)......Page 205
17.3 Splitting Fields (195)......Page 206
17.5 Separability (197)......Page 208
17.6 Galois Theory for Abstract Fields (202)......Page 213
Exercises (203)......Page 214
18.1 Transcendence Degree (205)......Page 216
18.2 Elementary Symmetric Polynomials (208)......Page 219
18.3 The General Polynomial (209)......Page 220
18.4 Cyclic Extensions (211)......Page 222
18.5 Solving Equations of Degree Four or Less (214)......Page 225
Exercises (218)......Page 229
19.1 Structure of Finite Fields (221)......Page 232
19.2 The Multiplicative Group (222)......Page 233
19.3 Application to Solitaire (224)......Page 235
Exercises (225)......Page 236
20.1 What Euclid Knew (227)......Page 238
20.2 Which Constructions are Possible? (230)......Page 241
20.3 Regular Polygons (231)......Page 242
20.5 How to Draw a Regular 17-gon (235)......Page 246
Exercises (240)......Page 251
21. Circle Division (243)......Page 253
21.1 Genuine Radicals (244)......Page 254
21.2 Fifth Roots Revisited (246)......Page 256
21.3 Vandermonde Revisited (249)......Page 259
21.4 The General Case (250)......Page 260
21.5 Cyclotomic Polynomials (253)......Page 263
21.6 Galois Group of ℚ(ξ): ℚ (255)......Page 265
21.7 The Technical Lemma (256)......Page 266
21.8 More on Cyclotomic Polynomials (257)......Page 267
21.9 Constructions Using a Trisector (259)......Page 269
Exercises (263)......Page 273
22.1 Transitive Subgroups (267)......Page 277
22.2 Bare Hands on the Cubic (268)......Page 278
22.3 The Discriminant (271)......Page 281
22.4 General Algorithm for the Galois Group (272)......Page 282
Exercises (274)......Page 284
23.1 Ordered Fields and Their Extensions (277)......Page 287
23.2 Sylow's Theorem (279)......Page 289
23.3 The Algebraic Proof (281)......Page 291
Exercises (282)......Page 292
24. Transcendental Numbers (285)......Page 294
24.1 Irrationality (286)......Page 295
24.2 Transcendence of e (288)......Page 297
24.3 Transcendence of π (289)......Page 298
Exercises (292)......Page 301
25. What Did Galois Do or Know? (295)......Page 303
25.2 The First Memoir (296)......Page 304
25.3 What Galois Proved (297)......Page 305
25.4 What is Galois Up To? 299)......Page 307
25.5 Alternating Groups, Especially ��₅ (301)......Page 309
25.6 Simple Groups Known to Galois (302)......Page 310
25.7 Speculations about Proofs (303)......Page 311
Exercises (307)......Page 315
Galois Theory (309)......Page 317
Additional Mathematical Material (310)......Page 318
Historical Material (312)......Page 320
The Internet (313)......Page 321
iPad App (314)......Page 322