Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of Q based on the existence or not existence of non-trivial 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 $Q\sqrt{d}$ has or not non-trivial solutions.

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 non-trivial 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 non-trivial solutions.

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

Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of Q based on the existence or not existence of non-trivial 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 $Q\sqrt{d}$ has or not non-trivial solutions.