Galois Theories 1st Edition by Francis Borceux, George Janelidze ISBN 978-0521803090 0521803098
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Product details:
ISBN 10: 0521803098
ISBN 13: 978-0521803090
Author: Francis Borceux, George Janelidze
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.
Table of contents:
1 Classical Galois theory
1.1 Algebraic extensions
1.2 Separable extensions
1.3 Normal extensions
1.4 Galois extensions
2 Galois theory of Grothendieck
2.1 Algebras on a field
2.2 Extension of scalars
2.3 Split algebras
2.4 The Galois equivalence
3 Infinitary Galois theory
3.1 Finitary Galois subextensions
3.2 Infinitary Galois groups
3.3 Classical infinitary Galois theory
3.4 Profinite topological spaces
3.5 Infinitary extension of the Galois theory of Grothendieck
4 Categorical Galois theory of commutative rings
4.1 Stone duality
4.2 Pierce representation of a commutative ring
4.3 The adjoint of the ‘spectrum’ functor
4.4 Descent morphisms
4.5 Morphisms of Galois descent
4.6 Internal presheaves
4.7 The Galois theorem for rings
5 Categorical Galois theorem and factorization systems
5.1 The abstract categorical Galois theorem
5.2 Central extensions of groups
5.3 Factorization systems
5.4 Reflective factorization systems
5.5 Semi-exact reflections
5.6 Connected components of a space
5.7 Connected components of a compact Hausdorff space
5.8 The monotone-light factorization
6 Covering maps
6.1 Categories of abstract families
6.2 Some limits in Fam(A)
6.3 Involving extensivity
6.4 Local connectedness and étale maps
6.5 Localization and covering morphisms
6.6 Classification of coverings
6.7 The Chevalley fundamental group
6.8 Path and simply connected spaces
7 Non-galoisian Galois theory
7.1 Internal presheaves on an internal groupoid
7.2 Internal precategories and their presheaves
7.3 A factorization system for functors
7.4 Generalized descent theory
7.5 Generalized Galois theory
7.6 Classical Galois theories
7.7 Grothendieck toposes
7.8 Geometric morphisms
7.9 Two dimensional category theory
7.10 The Joyal-Tierney theorem
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Tags: Francis Borceux, George Janelidze, Galois Theories