The set of nonsquares in a number field is diophantine

• Autores: Bjorn Poonen
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 16, Nº 1, 2009, págs. 165-170
• Idioma: inglés
• DOI: 10.4310/mrl.2009.v16.n1.a16
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• Resumen

  • 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 $y^2 - az^2 = P(x)$.