On a variant of the Beckmann--Black problem

TL;DR

This paper investigates a variant of the Beckmann--Black problem, demonstrating that every Galois extension can be realized as a specialization of a non-normal Galois extension, and explores automorphism groups over skew fields.

Contribution

It extends the Beckmann--Black problem by removing the normality requirement and shows all finite groups can appear as automorphism groups over certain skew fields.

Findings

Positive solution for non-normal Galois extensions with subgroup G

All finite groups occur as automorphism groups over specific skew fields

Results hold for arbitrary fields and groups, broadening applicability

Abstract

Given a field $k$ and a finite group $G$, the Beckmann--Black problem asks whether every Galois field extension $F/k$ with group $G$ is the specialization at some $t_0 \in k$ of some Galois field extension $E/k(T)$ with group $G$ and $E \cap \overline{k} = k$. We show that the answer is positive for arbitrary $k$ and $G$, if one waives the requirement that $E/k(T)$ is normal. In fact, our result holds if ${\rm{Gal}}(F/k)$ is any given subgroup $H$ of $G$ and, in the special case $H=G$, we provide a similar conclusion even if $F/k$ is not normal. We next derive that, given a division ring $H$ and an automorphism $\sigma$ of $H$ of finite order, all finite groups occur as automorphism groups over the skew field of fractions $H(T, \sigma)$ of the twisted polynomial ring $H[T, \sigma]$.