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

TL;DR

The paper proves that the set of non-squares in a number field is diophantine, using properties of conic bundles and Brauer-Manin obstructions, advancing understanding of diophantine sets in number theory.

Contribution

It establishes that the set of non-squares in a number field is diophantine, linking algebraic properties with diophantine definability through conic bundle analysis.

Findings

k* - k*^2 is diophantine over k

Finiteness of certain Brauer-Manin obstructions for conic bundles

Connection between non-squares and diophantine sets in number fields

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).