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

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 $k^{⟨\sigma ⟩}$ 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 $k^{⟨\sigma ⟩}$ 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 $H(t,\sigma )$.