Irreducibility in approximation couples

Abstract

To 80 th birthday of Paul ErdSsHilbert's irreducibility theorem and its validity for fields other than the field of rational numbers (such fields are called Hilbertian) is central in trying to realize some speciM finite groups as Galois groups. These Hilbertian fields are now being studied extensively.The property of approximation couples of rings arose out of Artin [1] on algebraic approximations. Recently, Robba [5] and Schoutens [6] gave examples of such couples.In this note we show that if A C .4 is a pair of domains having the property of approximation, then Hilbertianity of .~ implies Hilbertianity of A. To make this note self-contained, we recall the necessary definitions of pairs of rings with approximation property and Hilbertian rings and fields. We also give some known examples. In the last section, the main result is proved. w 1. Approximation CouplesLet A C -4 be a pair of commutative rings with the same identity and A be provided with a linear topology r which stems from a filtration {P2~} of ideals such that (i) N92n = 0 and (ii) A is dense in A.