On finite embedding problems with abelian kernels

TL;DR

This paper proves that certain finite split embedding problems with abelian kernels over Hilbertian fields have solutions that split specified valuations, and applies this to solve variants of the Beckmann-Black problem and inverse Galois problems over division rings.

Contribution

It establishes solutions to finite split embedding problems with abelian kernels over Hilbertian fields, with applications to Galois theory over fields and division rings.

Findings

Solutions to embedding problems with abelian kernels exist under valuation splitting conditions.

Every solvable Galois extension can be realized as a specialization of a Galois extension over a rational function field.

All finite semiabelian groups occur as Galois groups over certain skew fields.

Abstract

Given a Hilbertian field $k$ and a finite set $\mathcal{S}$ of Krull valuations of $k$, we show that every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/k)$ over $k$ with abelian kernel has a solu\-tion ${\rm{Gal}}(F/k) \rightarrow G$ such that every $v \in \mathcal{S}$ is totally split in $F/L$. Two applications are then given. Firstly, we solve a non-constant variant of the Beckmann--Black problem for solvable groups: given a field $k$ and a non-trivial finite solvable group $G$, every Galois field extension $F/k$ of group $G$ is shown to occur as the specialization at some $t_0 \in k$ of some Galois field extension $E/k(T)$ of group $G$ with $E \not \subseteq \overline{k}(T)$. Secondly, we contribute to inverse Galois theory over division rings, by showing that, for every division ring $H$ and every automorphism $\sigma$ of $H$ of finite order, all finite semiabelian groups occur as Galois groups over the skew field of fractions $H(T, \sigma)$ of the twisted polynomial ring $H[T, \sigma]$.