On the characterization of Hilbertian fields

TL;DR

This paper investigates conditions under which a field can be considered Hilbertian, showing that the irreducible specialization property for absolutely irreducible polynomials is sufficient, thus relaxing previous criteria.

Contribution

It establishes a relaxed criterion for Hilbertianity, focusing on absolutely irreducible polynomials, which broadens the understanding of Hilbertian fields.

Findings

Proves that the irreducible specialization property for absolutely irreducible polynomials suffices for Hilbertianity.

Provides a new characterization of Hilbertian fields.

Answers a question posed by D`ebes and Haran regarding Hilbertianity conditions.

Abstract

The main goal of this work is to answer a question of P. D`ebes and D. Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials.