Let $\omega=\frac{1}{2}(-1+\sqrt{-3})$.

(a) Prove that $\mathbb{Z}[\omega]$ is a Euclidean domain.

(b) Deduce that $\mathbb{Z}[\omega]$ is a unique factorisation domain, stating carefully any results from the course that you use.

(c) By working in $\mathbb{Z}[\omega]$, show that whenever $x, y \in \mathbb{Z}$ satisfy

$$x^{2}-x+1=y^{3}$$

then $x$ is not congruent to 2 modulo 3 .