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

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

2007-12-11

arxiv.org/abs/0712.1785v2

Mathematics

Number Theory

5 pages; corrected minor typos, improved exposition, added reference

Scientific paper

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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