Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

Abstract

Let K be an algebraic number field and IK the ring of algebraic integers in K. *K and *IK denote enlargements of K and IK respectively. Let x Є *K – K. In this paper, we are concerned with algebraic extensions of K(x) within *K. For each x Є *K – K and each natural number d, YK(x,d) is defined to be the number of algebraic extensions of K(x) of degree d within *K. x Є *K – K is called a Hilbertian element if YK(x,d) = 0 for all d Є N, d > 1; in other words, K(x) has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural number ω, 2ωPω and 2ω(ω3 + 1) are Hilbertian elements in *Q, where pω is the ωth prime number.