Galois closure data for extensions of rings

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of $\mathrm{S}_n$. A $G$-closure datum for $A$ over $R$ is an $R$-algebra homomorphism $\varphi: (A^{\otimes n})^{G}\to R$ satisfying certain properties, and we associate to a closure datum $\varphi$ a closure algebra $A^{\otimes n}\otimes_{(A^{\otimes n})^G} R$. This construction reproduces the normal closure of a finite separable field extension if $G$ is the corresponding Galois group. We describe G-closure data and algebras of finite \'etale algebras over a general connected ring $R$ in terms of the corresponding finite sets with continuous actions by the \'etale fundamental group of $R$. We show that if $2$ is invertible, then $\mathrm{A}_n$-closure data for free extensions correspond to square roots of the discriminant, and that $\mathrm{D}_4$-closure data for quartic monogenic extensions correspond to roots of the cubic resolvent.