Hilbertian field

A field $K$ is Hilbertian if each finite family of polynomials irreducible over $K(T)$ and separable in its root variable remains irreducible of the same degree at infinitely many $T=a\in K$, after any finitely many prescribed bad parameter values are excluded. Equivalently, every finite separable extension of $K(T)$ has infinitely many unramified $K$-rational places whose residue extension retains its full degree.

The rational numbers, and every number field, are Hilbertian fields. The classical integral form supplies infinitely many integral specializations avoiding a prescribed finite exceptional set. The theorem preserves irreducibility, rather than asserting that every specialization works: $X^2-T$ becomes reducible at square parameters.

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.

For a finite family of irreducible separable polynomials $f_i(T,X)$ and a nonzero exclusion polynomial $h(T)$, this is the set of parameters $a\in K$ with $h(a)\ne 0$ for which every specialization remains irreducible of its original degree. A Hilbertian field has infinitely many points in each such set. Denominators, leading coefficients and discriminants can be included in $h$.