Hilbertian field

Remarks

If $K$ is Hilbertian, then for such data there are infinitely many $a \in K$ with these properties, since one can add to $g$ factors $T - a$ for the values already found. A Hilbertian field is infinite, and the same condition then holds with $T$ replaced by several variables $T_1, \dots, T_r$. In characteristic $0$, the separability condition is automatic, and $K$ is Hilbertian exactly when $K$ itself is not a thin subset of the affine line over $K$.

Example: $\Q$ is Hilbertian by Hilbert's irreducibility theorem, and so is every number field. For any field $k$, the rational function field $k(t)$ is Hilbertian.

Non-example: $\R$ is not Hilbertian. The polynomial $f = X^2 - T^2 - 1$ is irreducible in $\R[T, X]$, because $T^2 + 1$ is not a square in $\R(T)$. But for every $a \in \R$ one has $a^2 + 1 > 0$, so $f(a, X) = (X - \sqrt{a^2 + 1})(X + \sqrt{a^2 + 1})$ factors over $\R$. An algebraically closed field is not Hilbertian either, because every polynomial in $X$ of degree at least $2$ factors over it.