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arXiv · 1601.07829

Irreducibility of Polynomials over Global Fields is Diophantine

Philip Dittmann
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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 any root in $K$. This strengthens the known result that the set of non-$n$-th-powers in $K$ is diophantine when $K$ is a number field. 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 $\mathbb{Q}$.

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Philip Dittmann. 2016-01-28. Irreducibility of Polynomials over Global Fields is Diophantine. https://doi.org/10.1112/s0010437x17007977

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