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Irreducibility of polynomials over global fields is diophantine

Part of: Number theory: Connections with logic Algebraic number theory: global fields

Published online by Cambridge University Press:  08 March 2018

Philip Dittmann
Philip Dittmann*
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK email dittmann@maths.ox.ac.uk
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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 $\mathbb{Q}$ .

Keywords

central simple algebradefinabilitydiophantine set

MSC classification

Primary: 11U05: Decidability
Secondary: 11R52: Quaternion and other division algebras

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 4 , April 2018 , pp. 761 - 772
Copyright
© The Author 2018 

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