By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field $K$ over which the regular inverse Galois problem can be shown to be solvable, but such that $K$ does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.

A (monic) polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ is called intersective if the congruence $f\left(x\right)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f\left(x\right)$nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over $\mathbb{Q}$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary...