Galois Theory

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Index is missing. 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