Galois-preserving Hilbert specialization

For a Finite Galois extension $E/K(T)$, take a primitive element and exclude all ramification and denominator problems. A Hilbert subset keeping its minimal polynomial irreducible has residue degree $[E:K(T)]$. The residue group is an unramified decomposition subgroup of the generic Galois group with the same order, so it is the whole group. This specializes a regular Galois realization over $K(T)$ to realizations over $K$ when $K$ is Hilbertian.