Inverse Galois problem

The inverse Galois problem asks which finite groups occur as Galois groups over a given field, with the classical problem taking the base field to be $\mathbb{Q}$. A regular realization over $K(T)$ and Galois-preserving Hilbert specialization solve an instance over a Hilbertian field. Realization over an algebraically closed constant field's rational function field alone does not supply descent to $\mathbb{Q}$.

Fix ordered branch positions and conjugacy classes $C_i$ in a finite group $G$. The generating product-one tuples $(g_1,\dots ,g_r)\in \prod C_i$ form the inner Nielsen set. The type is rigid when this set is nonempty and a single simultaneous-conjugation orbit. Over $\mathbb{C}(T)$ this fixes the connected Galois cover up to isomorphism: tuples prescribe the monodromy epimorphism of the punctured sphere, and conjugate epimorphisms give isomorphic covers; the local completions at punctures are uniquely determined by their cyclic monodromy.