Galois Closures for Rings

Abstract

To generalize the notion of Galois closure for separable field extensions, we devise a notion of G-closure for algebras of commutative rings R → A, where A is locally free of rank n as an R-module and G is a subgroup of Sn. A G-closure of A over R is an A⊗n-algebra B equipped with an R-algebra homomorphism (A⊗n)G → R satisfying certain properties. Being a G-closure commutes with base change, and reduces to being the normal closure of a finite separable field extension if G is the corresponding Galois group. We describe G-closures of finite étale algebras over connected rings in terms of the corresponding finite sets with continuous actions by the fundamental group. If 2 is invertible, then An-closures of free extensions correspond to square roots of the discriminant, and if 2 is a non-zerodivisor, then D4-closures of quartic monogenic extensions correspond to roots of the resolvent cubic.