Galois Theories

Develops Galois theory in a more general context, emphasizing category theory.

Francis Borceux (Author), George Janelidze (Author)

9780521803090, Cambridge University Press

Hardback, published 22 February 2001

356 pages
23.7 x 15.7 x 2.4 cm, 0.617 kg

Review of the hardback: 'A comprehensive account, which may well deepen one's understanding of the classical case.' Mathematika

Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context, presenting work by Grothendieck in terms of separable algebras and then proceeding to the infinite-dimensional case, which requires considering topological Galois groups. In the core of the book, 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. The first chapters are accessible to advanced undergraduates, with later ones at a graduate level. For all algebraists and category theorists this book will be a rewarding read.

Introduction
1. Classical Galois theory
2. Galois theory of Grothendieck
3. Infinitary Galois theory
4. Categorical Galois theory of commutative rings
5. Categorical Galois theorem and factorization systems
6. Covering maps
7. Non-Galoisian Galois theory
Appendix
Bibliography
Index.

Subject Areas: Algebra [PBF]