Irreducibility of polynomials over global fields is diophantine

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 a root in $K$. This strengthens a result by Colliot-Thélène and Van Geel [Compositio Math. 151 (2015), 1965–1980] stating that the set of non-$n$th powers in a number field $K$ is diophantine. 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 $\mathbb{Q}$.