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arXiv · 0712.1785

The set of non-squares in a number field is diophantine

Bjorn Poonen
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Abstract

Fix a number field k. We prove that k* - k*^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* 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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Bjorn Poonen. 2007-12-11. The set of non-squares in a number field is diophantine. https://doi.org/10.4310/mrl.2009.v16.n1.a16

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