Galois Closures for Rings

Abstract

To generalize the notion of Galois closure for separable field extensions, we devise a notion of <italic>G</italic>-closure for algebras of commutative rings <italic>R &rarr; A</italic>, where A is locally free of rank <italic>n</italic> as an <italic>R</italic>-module and <italic>G</italic> is a subgroup of <italic>Sn</italic>. A <italic>G</italic>-closure of <italic>A</italic> over <italic>R</italic> is an <italic>A<super>&#8855;n</super></italic>-algebra <italic>B</italic> equipped with an <italic>R</italic>-algebra homomorphism <italic>(A<super>&#8855;n</super>)<super>G</super> &rarr; R</italic> satisfying certain properties. Being a <italic>G</italic>-closure commutes with base change, and reduces to being the normal closure of a finite separable field extension if <italic>G</italic> is the corresponding Galois group. We describe <italic>G</italic>-closures of finite &eacute;tale algebras over connected rings in terms of the corresponding finite sets with continuous actions by the fundamental group. If 2 is invertible, then <italic>An</italic>-closures of free extensions correspond to square roots of the discriminant, and if 2 is a non-zerodivisor, then <italic>D</italic>4-closures of quartic monogenic extensions correspond to roots of the resolvent cubic.