Hilbertian field

Summary

A field is Hilbertian if irreducible polynomials in several variables stay irreducible after suitable values in the field are substituted for all but one variable. Hilbert's irreducibility theorem says that the field $\Q$ of rational numbers is Hilbertian. Hilbertian fields are the natural setting for realizing finite groups as Galois groups by specialization, an approach to the inverse Galois problem. In characteristic $0$, a field is Hilbertian exactly when its affine line is not a thin set in the sense of Serre.