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.

Toma Albu is Professor of Mathematics at Atilim University, Ankara, Turkey, and Bucharest University, Romania. His research interests involve ring theory, module theory, field theory, and algebraic number theory. Dr. Albu has authored or coauthored several books and more than 75 articles appearing in various international journals. He received the M.Sc. (1966) and Ph.D (1971) degrees from Bucharest University, Romania. He was a Humboldt Research Fellow at the Universities of Munich and Dusseldorf. Dr. Albu has also held Visiting Professor positions in Osaka, Padua, Milwaukee, Columbus, and Santa Barbara.

Preface

v

Introduction

1

(12)

Part 1. FINITE COGALOIS THEORY

13

(244)

Preliminaries

15

(38)

General notation and terminology

15

(4)

A short review of basic Field Theory

19

(20)

The Vahlen-Capelli Criterion

39

(8)

Bounded Abelian groups

47

(3)

Exercises to Chapter 1

50

(2)

Bibliographical comments to Chapter 1

52

(1)

Kneser Extensions

53

(16)

G-Radical and G-Kneser extensions

53

(7)

The Kneser Criterion

60

(5)

Exercises to Chapter 2

65

(2)

Bibliographical comments to Chapter 2

67

(2)

Cogalois Extensions

69

(20)

The Greither-Harrison Criterion

69

(5)

Examples and properties of Cogalois extensions

74

(9)

The Cogalois group of a quadratic extension

83

(3)

Exercises to Chapter 3

86

(2)

Bibliographical comments to Chapter 3

88

(1)

Strongly Kneser Extensions

89

(36)

Galois and Cogalois connections

90

(4)

Strongly G-Kneser extensions

94

(6)

G-Cogalois extensions

100

(4)

The Kneser group of a G-Cogalois extension

104

(4)

Almost G-Cogalois extensions

108

(12)

Exercises to Chapter 4

120

(3)

Bibliographical comments to Chapter 4

123

(2)

Galois G-Cogalois Extensions

125

(28)

Galois G-radical extensions

125

(3)

Abelian G-Cogalois extensions

128

(2)

Applications to elementary Field Arithmetic (I)

130

(18)

Exercises to Chapter 5

148

(3)

Bibliographical comments to Chapter 5

151

(2)

Radical Extensions and Crossed Homomorphisms

153

(20)

Galois extensions and crossed homomorphisms

154

(5)

Radical extensions via crossed homomorphisms

159

(7)

Exercises to Chapter 6

166

(5)

Bibliographical comments to Chapter 6

171

(2)

Examples of G-Cogalois Extensions

173

(18)

Classical Kummer extensions

173

(5)

Generalized Kummer extensions

178

(2)

Kummer extensions with few roots of unity

180

(1)

Quasi-Kummer extensions

181

(3)

Cogalois extensions

184

(2)

Exercises to Chapter 7

186

(3)

Bibliographical comments to Chapter 7

189

(2)

G-Cogalois Extensions and Primitive Elements

191

(16)

Primitive elements for G-Cogalois extensions

191

(5)

Applications to elementary Field Arithmetic (II)

196

(8)

Exercises to Chapter 8

204

(1)

Bibliographical comments to Chapter 8

205

(2)

Applications to Algebraic Number Fields

207

(22)

Number theoretic preliminaries

207

(5)

Some classical results via Cogalois Theory

212

(6)

Hecke systems of ideal numbers

218

(7)

Exercises to Chapter 9

225

(2)

Bibliographical comments to Chapter 9

227

(2)

Connections with Graded Algebras and Hopf Algebras

229

(28)

G-Cogalois extensions via strongly graded fields

229

(13)

Cogalois extensions and Hopf algebras

242

(11)

Exercises to Chapter 10

253

(2)

Bibliographical comments to Chapter 10

255

(2)

Part 2. INFINITE COGALOIS EXTENSIONS

257

(72)

Infinite Kneser Extensions

259

(10)

Infinite G-Kneser extensions

259

(3)

Infinite strongly Kneser extensions

262

(4)

Exercises to Chapter 11

266

(1)

Bibliographical comments to Chapter 11

267

(2)

Infinite G-Cogalois Extensions

269

(14)

The General Purity Criterion and its applications

269

(7)

Infinite Cogalois extensions

276

(3)

Exercises to Chapter 12

279

(2)

Bibliographical comments to Chapter 12

281

(2)

Infinite Kummer Theory

283

(8)

Infinite classical Kummer extensions

283

(2)

Infinite generalized Kummer extensions

285

(1)

Infinite Kummer extensions with few roots of unity

286

(1)

Infinite quasi-Kummer extensions

287

(2)

Exercises to Chapter 13

289

(1)

Bibliographical comments to Chapter 13

289

(2)

Infinite Galois Theory and Pontryagin Duality

291

(14)

Profinite groups and Infinite Galois Theory

291

(5)

Character group and Pontryagin Duality

296

(4)

Exercises to Chapter 14

300

(3)

Bibliographical comments to Chapter 14

303

(2)

Infinite Galois G-Cogalois Extensions

305

(24)

The infinite Kneser group via crossed homomorphisms

306

(8)

Lattice-isomorphic groups

314

(3)

Infinite Abelian G-Cogalois extensions

317

(8)

Exercises to Chapter 15

325

(2)

Bibliographical comments to Chapter 15

327

(2)

Bibliography

329

(6)

Index

335

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.