Probl\`emes de plongement finis sur les corps non commutatifs

TL;DR

This paper extends finite embedding problems from classical fields to non-commutative skew fields, providing solutions under certain conditions and generalizing inverse Galois theory results to non-commutative settings.

Contribution

It introduces a framework for solving finite embedding problems over skew fields and their fractions, extending inverse Galois theory to non-commutative algebra.

Findings

Finite embedding problems over skew fields relate to those over their centers with polynomial constraints.

Every constant finite split embedding problem over $H(t)$ has a solution if $h$ is ample.

Solutions are extended to skew fields of fractions of twisted polynomial rings with automorphisms of finite order.

Abstract

We extend finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of a skew field $H$ of finite dimension over its center $h$. First, we show that solving a finite embedding problem over $H$ is equivalent to finding a solution to some finite embedding problem over $h$ fulfilling a polynomial constraint. Next, we show that every constant finite split embedding problem over the skew field of fractions $H(t)$ with central indeterminate $t$ has a solution, if $h$ is an ample field. This is a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over the skew field of fractions $H(t, \sigma)$ of the twisted polynomial ring $H[t, \sigma]$, for some automorphisms $\sigma$ of $H$ of finite order. Our results extend previous works on the inverse Galois problem over skew fields.