Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of nontrivial solutions of Fermat's cubic equation in the ring of integers belonging to those extensions? Considering d as the only number necessary to determine if $\mathbb{Q}[\sqrt{d}]$ has or not nontrivial solutions.

$\begingroup$ See section 2 of this paper. $\endgroup$ – Wojowu Feb 26 at 20:38

$\begingroup$ I've already checked that paper. The result described, in general case, is not unknown for me. However I was not able to describe my idea clearly: i) Nontrivial solutions are searched in $Z[\omega _d]$, the ring of integers of $Q(\sqrt{d})$ $\endgroup$ – E. Pech Feb 26 at 23:41

$\begingroup$ ii) Criteria should be as simple as a modular congruence over Z. For example, "if $d\equiv 3(4)$ then there exist nontrivial solutions" Maybe I just can not see it, but I don't think Section 2 accomplish any of these points. $\endgroup$ – E. Pech Feb 26 at 23:49

$\begingroup$ Than you for all your help. $\endgroup$ – E. Pech Feb 26 at 23:50
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 SwinnertonDyer 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 nonexistence 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.

3$\begingroup$ (though  as with Tunnell's criterion  in one direction part of the result is still conditional on the conjectural BSD criterion for positive rank.) $\endgroup$ – Noam D. Elkies Feb 27 at 1:22