I don't want to toot my own horn, but I wrote a paper on this topic with a graduate student. Our result is conditional on the Birch and Swinnerton-Dyer conjecture. Basically, the Fermat cubic $x^3 + y^3 = z^3$ is isomorphic to $E : y^2 + y = x^3 - 7$. Since this curve has rank zero, the existence or non-existence of solutions in $\mathbb{Q}(\sqrt{d})$ is equivalent to the quadratic twist of $E_{d}$ having positive rank or not. One can then give a criterion for this by relating $L(E_{d},1)$ to the Fourier coefficients of weight $3/2$ modular forms. The method is very similar to Tunnell's solution of the congruent number problem.

I don't want to toot my own horn, but I coauthored a paper on this topic with Marvin Jones. One direction of our result is conditional on the Birch and Swinnerton-Dyer conjecture (see the first remark on page 3). Basically, the Fermat cubic $x^3 + y^3 = z^3$ is isomorphic to $E : y^2 + y = x^3 - 7$. Since this curve has rank zero over $\mathbb{Q}$, the existence or non-existence of solutions in $\mathbb{Q}(\sqrt{d})$ is equivalent to the quadratic twist $E_{d}$ having positive rank or not over $\mathbb{Q}$. One can then give a criterion for this by relating $L(E_{d},1)$ to the Fourier coefficients of weight $3/2$ modular forms. The method is very similar to Tunnell's solution of the congruent number problem.

I don't want to toot my own horn, but I wrote a paper on this topic with a graduate student. Our result is conditional on the Birch and Swinnerton-Dyer conjecture. Basically, the Fermat cubic $x^3 + y^3 = z^3$ is isomorphic to $E : y^2 + y = x^3 - 7$. Since this curve has rank zero, the existence or non-existence of solutions in $\mathbb{Q}(\sqrt{d})$ is equivalent to the quadratic twist of $E_{d}$ having positive rank or not. One can then give a criteria for this by relating $L(E_{d},1)$ to the Fourier coefficients of weight $3/2$ modular forms. The method is very similar to Tunnell's solution of the congruent number problem.

I don't want to toot my own horn, but I wrote a paper on this topic with a graduate student. Our result is conditional on the Birch and Swinnerton-Dyer conjecture. Basically, the Fermat cubic $x^3 + y^3 = z^3$ is isomorphic to $E : y^2 + y = x^3 - 7$. Since this curve has rank zero, the existence or non-existence of solutions in $\mathbb{Q}(\sqrt{d})$ is equivalent to the quadratic twist of $E_{d}$ having positive rank or not. One can then give a criterion for this by relating $L(E_{d},1)$ to the Fourier coefficients of weight $3/2$ modular forms. The method is very similar to Tunnell's solution of the congruent number problem.