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

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).