Show that the ring $\mathbf{Z}[\omega]$ is Euclidean, where $\omega=\exp (2 \pi i / 3)$.

Show that a prime number $p \neq 3$ is reducible in $\mathbf{Z}[\omega]$ if and only if $p \equiv 1(\bmod 3)$.

Which prime numbers $p$ can be written in the form $p=a^{2}+a b+b^{2}$ with $a, b \in \mathbf{Z}$ (and why)?