Galois Theories of Fields and Rings

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.

“This book is based on a master’s course in Galois theory held at the University of Coimbra (Portugal) in 2003. … it is a clearly written book in Landau style suitable for graduate students, and it presents material not easily found elsewhere.” (Franz Lemmermeyer, zbMATH 1554.12001, 2025)

• 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.