Irreducibility and embedding problems

TL;DR

This paper investigates the conditions under which irreducible specializations and group-preserving specializations exist, providing a criterion based on embedding problems, with applications to number theory and field theory.

Contribution

It introduces a new criterion for irreducibility and embedding problems, advancing understanding of specialization issues in algebraic structures.

Findings

Criterion for irreducible specializations via embedding problems

Examples related to Schinzel's hypothesis H

Applications to Hilbertian fields

Abstract

We study irreducible specializations, in particular when group-preserving specializations may not exist. We obtain a criterion in terms of embedding problems. We include several applications to analogs of Schinzel's hypothesis H and to the theory of Hilbertian fields.