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PREFACE v
INTRODUCTION 1
Part 1. FINITE COGALOIS THEORY 13
Chapter 1. PRELIMINARIES 15
1.1. General notation and terminology 15
1.2. A short review of basic Field Theory 19
1.3. The Vahlen-Capelli Criterion 39
1.4. Bounded Abelian groups 47
1.5. Exercises to Chapter 1 50
1.6. Bibliographical comments to Chapter 1 52
Chapter 2. KNESER EXTENSIONS 53
2.1. G-Radical and G-Kneser extensions 53
2.2. The Kneser Criterion 60
2.3. Exercises to Chapter 2 65
2.4. Bibliographical comments to Chapter 2 67
Chapter 3. COGALOIS EXTENSIONS 69
3.1. The Greither-Harrison Criterion 69
3.2. Examples and properties of Cogalois extensions 74
3.3. The Cogalois group of a quadratic extension 83
3.4. Exercises to Chapter 3 86
3.5. Bibliographical comments to Chapter 3 88
Chapter 4. STRONGLY KNESER EXTENSIONS 89
4.1. Galois and Cogalois connections 90
4.2. Strongly G-Kneser extensions 94
4.3. G-Cogalois extensions 100
4.4. The Kneser group of a G-Cogalois extension 104
4.5. Almost G-Cogalois extensions 108
4.6. Exercises to Chapter 4 120
4.7. Bibliographical comments to Chapter 4 123
Chapter 5. GALOIS G-COGALOIS EXTENSIONS 125
5.1. Galois G-radical extensions 125
5.2. Abelian G-Cogalois extensions 128
5.3. Applications to elementary Field Arithmetic (I) 130
5.4. Exercises to Chapter 5 148
5.5. Bibliographical comments to Chapter 5 151
Chapter 6. RADICAL EXTENSIONS AND CROSSED
HOMOMORPHISMS 153
6.1. Galois extensions and crossed homomorphisms 154
6.2. Radical extensions via crossed homomorphisms 159
6.3. Exercises to Chapter 6 166
6.4. Bibliographical comments to Chapter 6 171
Chapter 7. EXAMPLES OF G-COGALOIS EXTENSIONS 173
7.1. Classical Kummer extensions 173
7.2. Generalized Kummer extensions 178
7.3. Kummer extensions with few roots of unity 180
7.4. Quasi-Kummer extensions 181
7.5. Cogalois extensions 184
7.6. Exercises to Chapter 7 186
7.7. Bibliographical comments to Chapter 7 189
Chapter 8. G-COGALOIS EXTENSIONS AND
PRIMITIVE ELEMENTS 191
8.1. Primitive elements for G-Cogalois extensions 191
8.2. Applications to elementary Field Arithmetic (II) 196
8.3. Exercises to Chapter 8 204
8.4. Bibliographical comments to Chapter 8 205
Chapter 9. APPLICATIONS TO ALGEBRAIC
NUMBER FIELDS 207
9.1. Number theoretic preliminaries 207
9.2. Some classical results via Cogalois Theory 212
9.3. Hecke systems of ideal numbers 218
9.4. Exercises to Chapter 9 225
9.5. Bibliographical comments to Chapter 9 227
Chapter 10. CONNECTIONS WITH GRADED ALGEBRAS
AND HOPF ALGEBRAS 229
10.1. G-Cogalois extensions via strongly graded fields 229
10.2. Cogalois extensions and Hopf algebras 242
10.3. Exercises to Chapter 10 253
10.4. Bibliographical comments to Chapter 10 255
Part 2. INFINITE COGALOIS THEORY 257
Chapter 11. INFINITE KNESER EXTENSIONS 259
11.1. Infinite G-Kneser extensions 259
11.2. Infinite strongly Kneser extensions 262
11.3. Exercises to Chapter 11 266
11.4. Bibliographical comments to Chapter 11 267
Chapter 12. INFINITE G-COGALOIS EXTENSIONS 269
12.1. The General Purity Criterion and its applications 269
12.2. Infinite Cogalois extensions 276
12.3. Exercises to Chapter 12 279
12.4. Bibliographical comments to Chapter 12 281
Chapter 13. INFINITE KUMMER THEORY 283
13.1. Infinite classical Kummer extensions 283
13.2. Infinite generalized Kummer extensions 285
13.3. Infinite Kummer extensions with few roots of unity 286
13.4. Infinite quasi-Kummer extensions 287
13.5. Exercises to Chapter 13 289
13.6. Bibliographical comments to Chapter 13 289
Chapter 14. INFINITE GALOIS THEORY AND
PONTRYAGIN DUALITY 291
14.1. Profinite groups and Infinite Galois Theory 291
14.2. Character group and Pontryagin Duality 296
14.3. Exercises to Chapter 14 300
14.4. Bibliographical comments to Chapter 14 303
Chapter 15. INFINITE GALOIS G-COGALOIS
EXTENSIONS 305
15.1. The infinite Kneser group via crossed homomorphisms 306
15.2. Lattice-isomorphic groups 314
15.3. Infinite Abelian G-Cogalois extensions 317
15.4. Exercises to Chapter 15 325
15.5. Bibliographical comments to Chapter 15 327
Bibliography 329
Index 335
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