Mathematical Research Letters

Volume 16 (2009)

Number 1

The set of nonsquares in a number field is diophantine

Pages: 165 – 170

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n1.a16

Author

Bjorn Poonen (Massachusetts Institute of Technology)

Abstract

Fix a number field $k$. We prove that $k^\times - k^{\times 2}$ is diophantine over $k$. This is deduced from a theorem that for a nonconstant separable polynomial $P(x) \in k[x]$, there are at most finitely many $a \in k^\times$ modulo squares such that there is a Brauer-Manin obstruction to the Hasse principle for the conic bundle $X$ given by $y^2 - az^2 = P(x)$.

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