Cogalois Theory, Vol. 252 Book

	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