Galois Theories of Fields and Rings

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.

Historical introduction.- Part I Some Galois theorems for fields.- 1 The classical Galois theorem.- 2 The Galois theorem of Grothendieck.- 3 Profinite topological spaces.- 4 The Galois theorems in arbitrary dimension.- Part II The Galois theory of rings.- 5 Adjunctions and monads.- 6 Profinite groupoids and presheaves.- 7 The descent theory of rings.- 8 The Pierce spectrum of a ring.- 9 The Galois theorem for rings.- Further Reading.- Index.

Francis Borceux is a category theorist at the University of Louvain, Belgium. He has developed research in algebra and essentially taught geometry, number theory, and algebra courses.