Cogalois Theory 1st Edition by Toma Albu – Ebook PDF Instant Download/Delivery: 1482275937, 9781482275933
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ISBN 10: 1482275937
ISBN 13: 9781482275933
Author: Toma Albu
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 correspondence. Solidly backed by over 250 exercises and an extensive bibliography, this book presents a compact and complete review of basic field theory, considers the Vahlen-Capelli Criterion, investigates the radical, Kneser, strongly Kneser, Cogalois, and G-Cogalois extensions, discusses field extensions that are simultaneously Galois and G-Cogalois, and presents nice applications to elementary field arithmetic.
Cogalois Theory 1st Table of contents:
Part 1. Finite Cogalois Theory
Chapter 1: Preliminaries
1.1. General notation and terminology
1.2. A short review of basic Field Theory
1.3. The Vahlen-Capelli Criterion
1.4. Bounded Abelian groups
1.5. Exercises to Chapter 1
1.6. Bibliographical comments to Chapter 1
Chapter 2: Kneser Extensions
2.1. G-Radical and G-Kneser extensions
2.2. The Kneser Criterion
2.3. Exercises to Chapter 2
2.4. Bibliographical comments to Chapter 2
Chapter 3: Cogalois Extensions
3.1. The Greither-Harrison Criterion
3.2. Examples and properties of Cogalois extensions
3.3. The Cogalois group of a quadratic extension
3.4. Exercises to Chapter 3
3.5. Bibliographical comments to Chapter 3
Chapter 4: Strongly Kneser Extensions
4.1. Galois and Cogalois connections
4.2. Strongly G-Kneser extensions
4.3. G-Cogalois extensions
4.4. The Kneser group of a G-Cogalois extension
4.5. Almost G-Cogalois extensions
4.6. Exercises to Chapter 4
4.7. Bibliographical comments to Chapter 4
Chapter 5: Galois G-Cogalois Extensions
5.1. Galois G-radical extensions
5.2. Abelian G-Cogalois extensions
5.3. Applications to elementary Field Arithmetic (I)
5.4. Exercises to Chapter 5
5.5. Bibliographical comments to Chapter 5
Chapter 6: Radical Extensions and Crossed Homomorphisms
6.1. Galois extensions and crossed homomorphisms
6.2. Radical extensions via crossed homomorphisms
6.3. Exercises to Chapter 6
6.4. Bibliographical comments to Chapter 6
Chapter 7: Examples of G-Cogalois Extensions
7.1. Classical Kummer extensions
7.2. Generalized Kummer extensions
7.3. Kummer extensions with few roots of unity
7.4. Quasi-Kummer extensions
7.5. Cogalois extensions
7.6. Exercises to Chapter 7
7.7. Bibliographical comments to Chapter 7
Chapter 8: G-Cogalois Extensions and Primitive Elements
8.1. Primitive elements for G-Cogalois extensions
8.2. Applications to elementary Field Arithmetic (II)
8.3. Exercises to Chapter 8
8.4. Bibliographical comments to Chapter 8
Chapter 9: Applications to Algebraic Number Fields
9.1. Number theoretic preliminaries
9.2. Some classical results via Cogalois Theory
9.3. Hecke systems of ideal numbers
9.4. Exercises to Chapter 9
9.5. Bibliographical comments to Chapter 9
Chapter 10: Connections with Graded Algebras and Hopf Algebras
10.1. G-Cogalois extensions via strongly graded fields
10.2. Cogalois extensions and Hopf algebras
10.3. Exercises to Chapter 10
10.4. Bibliographical comments to Chapter 10
Part 2. Infinite Cogalois Theory
Chapter 11: Infinite Kneser Extensions
11.1. Infinite G-Rneser extensions
11.2. Infinite strongly Kneser extensions
11.3. Exercises to Chapter 11
11.4. Bibliographical comments to Chapter 11
Chapter 12: Infinite G-Cogalois Extensions
12.1. The General Purity Criterion and its applications
12.2. Infinite Cogalois extensions
12.3. Exercises to Chapter 12
12.4. Bibliographical comments to Chapter 12
Chapter 13: Infinite Kummer Theory
13.1. Infinite classical Kummer extensions
13.2. Infinite generalized Kummer extensions
13.3. Infinite Kummer extensions with few roots of unity
13.4. Infinite quasi-Kummer extensions
13.5. Exercises to Chapter 13
13.6. Bibliographical comments to Chapter 13
Chapter 14: Infinite Galois Theory and Pontryagin Duality
14.1. Profinite groups and Infinite Galois Theory
14.2. Character group and Pontryagin Duality
14.3. Exercises to Chapter 14
14.4. Bibliographical comments to Chapter 14
Chapter 15: Infinite Galois G-Cogalois Extensions
15.1. The infinite Kneser group via crossed homomorphisms
15.2. Lattice-isomorphic groups
15.3. Infinite Abelian G-Cogalois extensions
15.4. Exercises to Chapter 15
15.5. Bibliographical comments to Chapter 15
Bibliography
Index
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