The Other Group of as Galois Extension

TL;DR

This paper constructs a canonical subgroup scheme associated with a Galois extension, revealing that it is constant precisely when the Galois group is abelian, thus linking group scheme structure to classical Galois theory.

Contribution

It introduces a canonical subgroup scheme for Galois extensions, providing a new perspective on the relationship between Galois groups and automorphism group schemes.

Findings

The subgroup scheme is constant if and only if the Galois group is abelian.

When Galois group is abelian, the subgroup scheme coincides with the Galois group.

The construction offers a canonical way to associate group schemes to Galois extensions.

Abstract

Let $k\subseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $\mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $\underline{G}\subset\mathscr{G}$ with the property that $Spec_k(K)$ is a $k$-torsor for $\underline{G}$. $\underline{G}$ is a constant $k$-group if and only if $G$ is abelian, in which case $G=\underline{G}$.