Solutions of the cubic Fermat equation in quadratic fields

TL;DR

This paper establishes criteria, conditional on the Birch and Swinnerton-Dyer conjecture, for the existence of non-trivial solutions to the cubic Fermat equation within quadratic fields, linking to the congruent number problem.

Contribution

It provides necessary and sufficient conditions on squarefree integers for solutions in quadratic fields, extending the understanding of Fermat's equation in algebraic number theory.

Findings

Conditions are similar to Tunnell's criteria for the congruent number problem.

Solutions depend on the Birch and Swinnerton-Dyer conjecture.

Characterizes when cubic Fermat solutions exist in quadratic fields.

Abstract

We give necessary and sufficient conditions on a squarefree integer $d$ for there to be non-trivial solutions to $x^{3} + y^{3} = z^{3}$ in $\Q(\sqrt{d})$, conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.