On the congruent number problem over integers of cyclic number fields

Abstract

Abstract Given a cyclic field extension K/ℚ of degree d and a nonzero rational integer m, we show that the equation mp2 = x4 - y 2 has no nontrivial solutions in K / Q ${K/{\mathbb Q}}$ when p belongs to a subset of rational prime numbers of relative density at least φ(d)/(2d).