Submission date: 28 January 2016

Abstract:

Given a number 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. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of Z in Q.

Mathematics Subject Classification: 11U05 (Primary), 11R52 (Secondary)

Keywords and phrases:

Full text arXiv 1601.07829: pdf, ps.