On the equation \(x^3 + y^3 + z^3 = 0\) in the maximal orders of quadratic fields (Q2911183)

The problem of finding integer solutions of the title equation in a quadratic field is equivalent to finding integer points on certain ellpitic curve in the same quadratic field. The main theorem of the paper gives necessary conditions for the solvability of the cubic equation in the ring of integers of a quartic number field. It is shown that in \(\mathbb Q(\sqrt{D})\) the cubic equation has infinitely many integer solutions if and only if the elliptic curve \(Y^2=X^3-2(6D)^3\) has rank \(\geq 1\) over \(\mathbb Q\).