A (pure) field is Hilbertian if there is some elementary extension and an element such that is relatively algebraically closed in . (Note that must be transcendental over .)

Usually Hilbertianity is phrased as saying that some analog of Hilbert's irreducibility theorem holds. One version of this says that can't be covered by a finite union of sets of the form where is a variety over , is a -definable morphism of varieties, and where either or and 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 is a countable (perfect?) Hilbertian field, and is a random automorphism of , then the fixed field of will be pseudo-finite for all in a set of Haar measure 1.

Hilbertian fields form an elementary class. (right?) ::: ::: :::