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Galois Theories

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Overview

Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context. The authors first formalize the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience, the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit and adjoint functors. For all algebraists and category theorists this book will be a rewarding read.

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Editorial Reviews

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"This is a clearly written and readable book covering a lot of interesting and important material, and effectively leading the reader through increasing levels of generality and abstraction." Mathematical Review
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Product Details

Table of Contents

Preface
1 Classical Galois theory 1
1.1 Algebraic extensions 1
1.2 Separable extensions 4
1.3 Normal extensions 6
1.4 Galois extensions 8
2 Galois theory of Grothendieck 15
2.1 Algebras on a field 15
2.2 Extension of scalars 20
2.3 Split algebras 23
2.4 The Galois equivalence 27
3 Infinitary Galois theory 36
3.1 Finitary Galois subextensions 36
3.2 Infinitary Galois groups 39
3.3 Classical infinitary Galois theory 44
3.4 Profinite topological spaces 47
3.5 Infinitary extension of the Galois theory of Grothendieck 56
4 Categorical Galois theory of commutative rings 65
4.1 Stone duality 65
4.2 Pierce representation of a commutative ring 72
4.3 The adjoint of the 'spectrum' functor 80
4.4 Descent morphisms 91
4.5 Morphisms of Galois descent 98
4.6 Internal presheaves 102
4.7 The Galois theorem for rings 106
5 Categorical Galois theorem and factorization systems 116
5.1 The abstract categorical Galois theorem 117
5.2 Central extensions of groups 127
5.3 Factorization systems 144
5.4 Reflective factorization systems 149
5.5 Semi-exact reflections 156
5.6 Connected components of a space 168
5.7 Connected components of a compact Hausdorff space 170
5.8 The monotone-light factorization 177
6 Covering maps 186
6.1 Categories of abstract families 186
6.2 Some limits in Fam (A) 189
6.3 Involving extensivity 193
6.4 Local connectedness and etale maps 197
6.5 Localization and covering morphisms 201
6.6 Classification of coverings 207
6.7 The Chevalley fundamental group 212
6.8 Path and simply connected spaces 216
7 Non-galoisian Galois theory 225
7.1 Internal presheaves on an internal groupoid 225
7.2 Internal precategories and their presheaves 241
7.3 A factorization system for functors 246
7.4 Generalized descent theory 251
7.5 Generalized Galois theory 258
7.6 Classical Galois theories 261
7.7 Grothendieck toposes 266
7.8 Geometric morphisms 274
7.9 Two dimensional category theory 287
7.10 The Joyal-Tierney theorem 294
App: Final remarks 304
Bibliography 331
Index of symbols 336
General index 338
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