  Hilbert specialization is an important tool in Field Arithmetic and
Arithmetic Geometry, which has usually been intended for polynomials, hence
hypersurfaces, and at scalar values. In this article, first, we extend this
tool to prime ideals, hence affine varieties, and offer an application to the
study of the irreducibility of the intersection of varieties. Then, encouraged
by recent results, we consider the more general situation in which the
specialization is done at polynomial values, instead of scalar values.
