# Le probl\`eme inverse de Galois sur les corps des fractions tordus \`a ind\'etermin\'ee centrale

## TL;DR

This paper establishes an equivalence between the inverse Galois problem over certain skew fields and a polynomial-constrained variant over their centers, with implications for fields containing ample subfields.

## Contribution

It introduces a new equivalence linking the inverse Galois problem over skew fields to a polynomial-constrained problem over their centers, extending understanding in non-commutative algebra.

## Key findings

- Inverse Galois problem over skew fields reduces to a polynomial problem over centers
- Positive results for fields containing ample subfields
- Application to rational function fields with central indeterminate

## Abstract

In this article, we show that the Inverse Galois Problem over a skew field $H$ of finite dimension over its center $k$ is equivalent to a variant of the Inverse Galois Problem over $k$ involving a polynomial constraint. As an application, we show that if $k$ contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field $H(t)$ of rational fractions with central indeterminate.