Pre-Galois Theory

Abstract

We introduce and study a class of field extensions that we call $pre$-$Galois$; viz. extensions that become Galois after some linearly disjoint Galois base change $L/k$. Among them are $geometrically Galois$ extensions of $\kappa(T)$, with $\kappa$ a field: extensions that become Galois and remain of the same degree over $\overline \kappa(T)$. We develop a pre-Galois theory that includes a Galois correspondence, and investigate the corresponding variants of the inverse Galois problem. We provide answers in situations where the classical analogs are not known. In particular, for every finite simple group $G$, some power $G^n$ is a geometric Galois group over $k$, and is a pre-Galois group over $k$ if $k$ is Hilbertian. For every finite group $G$, the same conclusion holds for $G$ itself ($n=1$) if $k=\mathbb{Q}^{\rm ab}$ and $G$ has a weakly rigid tuple of conjugacy classes; and then $G$ is a regular Galois group over an extension of $\mathbb{Q}^{\rm ab}$ of degree dividing the order of Out$(G)$. We also show that the inverse problem for pre-Galois extensions over a field $k$ (that every finite group is a pre-Galois group over $k$) is equivalent to the $a$ $priori$ stronger inverse Galois problem over $k$, and similarly for the geometric vs. regular variants.