Irreducibility of Polynomials over Global Fields is Diophantine

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 $\mathbb{Q}$.

References in corpus (3)

• Le complémentaire des puissances $n$-ièmes dans un corps de nombres est un ensemble diophantien
• Universally and existentially definable subsets of global fields
• Integrality at a prime for global fields and the perfect closure of global fields of characteristic p>2

Cited by in corpus (4)

• A p-adic analogue of Siegel's Theorem on sums of squares
• Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers
• Universally and existentially definable subsets of global fields
• A class of fields with a restricted model completeness property