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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 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 $h$.

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 $\mathbb{Q}$.