Irreducibility of Polynomials over Global Fields is Diophantine

TL;DR

This paper develops a diophantine criterion for determining when polynomials over global fields lack roots in the field, extending known results and providing tools for analyzing polynomial irreducibility through a generalized quaternion method.

Contribution

It introduces a diophantine criterion for polynomial irreducibility over global fields, generalizing previous results and employing a novel quaternion-based approach.

Findings

Established a diophantine criterion for roots of polynomials over global fields.

Extended the diophantine characterization to polynomial irreducibility.

Utilized a generalized quaternion method for the analysis.

Abstract

Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers in $K$ is diophantine when $K$ is a number field. We also deduce a diophantine criterion for a polynomial over $K$ of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of $\mathbb{Z}$ in $\mathbb{Q}$.