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

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 $H(t,\sigma)$.