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Preprint Number 997

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997. Philip Dittmann
Not having a Root in Number Fields is Diophantine

Submission date: 28 January 2016


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)

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Full text arXiv 1601.07829: pdf, ps.

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