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Book Categories |
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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Add Cogalois Theory, Vol. 252, This volume offers a systematic, comprehensive investigation of field extensions, finite or not, that possess a Cogalois correspondence. The subject is somewhat dual to the very classical Galois Theory dealing with field extensions possessing a Galois cor, Cogalois Theory, Vol. 252 to the inventory that you are selling on WonderClubX
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Add Cogalois Theory, Vol. 252, This volume offers a systematic, comprehensive investigation of field extensions, finite or not, that possess a Cogalois correspondence. The subject is somewhat dual to the very classical Galois Theory dealing with field extensions possessing a Galois cor, Cogalois Theory, Vol. 252 to your collection on WonderClub |