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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 theInverse Galois Problem over $k$ involving a polynomial constraint. As anapplication, we show that if $k$ contains an ample field, then the InverseGalois Problem has a positive answer over the skew field $H(t)$ of rationalfractions with central indeterminate.

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