Schneider R. Convex Borceux F. Galois Theories of Fields and Rings 2024
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Schneider R. Convex Borceux F. Galois Theories of Fields and Rings 2024
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This textbook arises from a master’s course taught by the author at the University of Coimbra. It takes the reader from the very classical Galois theorem for fields to its generalization to the case of rings. Given a finite-dimensional Galois extension of fields, the classical bijection between the intermediate field extensions and the subgroups of the corresponding Galois group was extended by Grothendieck as an equivalence between finite-dimensional split algebras and finite sets on which the Galois group acts. Adding further profinite topologies on the Galois group and the sets on which it acts, these two theorems become valid in arbitrary dimension. Taking advantage of the power of category theory, the second part of the book generalizes this most general Galois theorem for fields to the case of commutative rings. This book should be of interest to field theorists and ring theorists wanting to discover new techniques which make it possible to liberate Galois theory from its traditional restricted context of field theory. It should also be of great interest to category theorists who want to apply their everyday techniques to produce deep results in other domains of mathematics.
Preface
Historical Introduction
Some Galois Theorems for Fields
The Classical Galois Theorem
The Galois Theorem of Grothendieck
Profinite Topological Spaces
The Galois Theorems in Arbitrary Dimension
Adjunctions and Monads
Profinite Groupoids and Presheaves
The Descent Theory of Rings
The Pierce Spectrum of a Ring
The Galois Theorem for Rings
Further Reading
Index