This is working in $\displaystyle Q[sqrt(-3)]$.

Let $\displaystyle H = (3 + sqrt(-3))/2$.

Assume $\displaystyle x, y, z$ are integers in $\displaystyle Q[sqrt(-3)]$.

If $\displaystyle z^3 = x^3 + y^3$, prove that $\displaystyle H$ divides $\displaystyle x, y, or z$.