The Galois closure for rings and some related constructions

Abstract

Let R be a ring and let A be a finite projective R-algebra of rank n. Manjul Bhargava and Matthew Satriano have recently constructed an R-algebra G(A/R), the Galois closure of A/R. Many natural questions were asked at the end of their paper. Here we address one of these questions, proving the existence of the natural constructions they call intermediate Sn-closures. We will also study properties of these constructions, generalizing some of their results, and proving new results both on the intermediate Sn-closures and on G(A/R).

Galois Closures of Non-commutative Rings and an Application to Hermitian Representations

“…, a (n) behave as if they are "Galois conjugates." One key property is that G(A/R) commutes with base change on R. This construction has since been generalized by Gioia to so-called intermediate Galois closure [Gio16] as well as by Biesel to Galois closures associated to subgroups of S n [Bie18].…”