In this paper we investigate the factorization of the polynomials in the special case where is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that is monic and linear.
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 over which the regular inverse Galois problem can be shown to be solvable, but such that does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.