A (pure) field K is Hilbertian if there is some elementary extension K* ≽ K and an element t ∈ K* ∖ K such that K(t) is relatively algebraically closed in K*. (Note that t must be transcendental over K.)
Usually Hilbertianity is phrased as saying that some analog of Hilbert's irreducibility theorem holds. One version of this says that Kn can't be covered by a finite union of sets of the form f(V(k)) where V is a variety over k, f : V → 𝔸n is a K-definable morphism of varieties, and where either dim V < n or dim V = n and V → 𝔸n is a dominant rational map of degree greater than 1.
Global fields are Hilbertian, as are function fields over arbitrary fields. In Fried and Jarden's book on Field Arithmetic, there is a theorem to the effect that any field having a suitable product formula is Hilbertian. Hilbertian fields play an important role in field arithmetic. There is some theorem, for example, which says that if K is a countable (perfect?) Hilbertian field, and σ is a random automorphism of Gal(Kalg/K), then the fixed field of σ will be pseudo-finite for all σ in a set of Haar measure 1.
Hilbertian fields form an elementary class. (right?) ::: ::: :::