Three variable, third degree Diophantine equation

The technique used to prove that $x^3 + y^3 + z^3 = 0$ has no non-trivial solutions in $\mathbb{Z}(\sqrt{-3})$ is also applicable to showing that $x^3 + y^3 = 3z^3$ has no non-trivial solutions (in $\mathbb{Z}(\sqrt{-3})$).

In fact, this appears (with proof) in section 13.5 as Theorem 232 in the excellent book, "An Introduction to the Theory of Numbers", by Hardy & Wright, 5th Edition (I have the Indian Edition, so might be a bit different from yours).

Here is a snapshot I managed to scrape (though the notation is quite old, and you would need parts of the rest of the book to make sense of it).