Solutions of the cubic Fermat equation in quadratic fields

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 $\text{Q}(\sqrt{d})$, 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.