Th\'eorie inverse de Galois sur les corps des fractions rationnelles tordus

TL;DR

This paper proves that the inverse Galois problem has a positive answer over certain skew fields of twisted rational fractions, especially when the fixed subfield contains ample, real closed, or Henselian fields with roots of unity.

Contribution

It establishes the inverse Galois problem for skew fields of twisted rational fractions under specific conditions on the fixed subfield, extending classical results to noncommutative settings.

Findings

Inverse Galois problem is solvable over $H(t,\sigma)$ when fixed subfield contains an ample field.

The profree group of countable rank is realizable as a Galois group over $H(t,\sigma)$ under certain conditions.

Conditions on the fixed subfield include containing a real closed or Henselian field with roots of unity.

Abstract

In this article, we prove that if $H$ is a skew field of center $k$ and $\sigma$ an automorphism of finite order of $H$ such that the fixed subfield $k^{\langle \sigma \rangle}$ of $k$ under the action of $\sigma$ contains an ample field, then the inverse Galois problem has a positive answer over the skew field $H(t,\sigma)$ of twisted rational fractions. Moreover, if $k^{\langle \sigma \rangle}$ contains either a real closed field, or an Henselian field of residue characteristic $0$ and containing all roots of unity, then the profree group of countable rank $\widehat{F}_{\omega}$ is a Galois group over $H(t,\sigma)$.