 Hilbert subset

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 $h$.