Un raffinement du caractère hilbertien du corps K(X)

Given an infinite field K. We will show that for the elements of a hilbertian set of K(X) associated to a family of polynomials with coefficients in K, the specialized polynomials remain also primitive knowing that the starting polynomials are it. This result refines the hilbertian character of the field K(X). Moreover, by an application of this result, one obtains an analog of the famous arithmetic progression theorem.