Irreducibility of polynomials over global fields is diophantine

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 Z in Q.