  We give a criterion to determine the existence of the nontrivial integral
solutions of the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by
constructing a relationship with the points on the elliptic curve
$y^2=x^3-432d^3k^2$ over $\mathbb{Q}$ for $k\in\mathbb{N}$.
