Let $R=\mathbb Z[e^{i\pi/3}]=\{a+be^{i\pi/3}\mid a,b\in \mathbb Z\}\subseteq \mathbb C$
(a) Show that $R$ is a Euclidean domain using the Euclidean norm $N(u)=|u|^2$.
(b) Show that if $p$ is a prime number in $\mathbb Z^+$ and $x,y\in \mathbb Z$ with $x^2+xy+y^2=p$, then $x+ye^{i\pi/3}$ is prime in $R$.
(c) Show that if $p$ is a prime number in $\mathbb Z^+$ and $p\neq x^2+xy+y^2$ for any $x,y\in\mathbb Z$, then $p$ is prime in $R$.
I don't know how to show that the division algorithm holds in $R$.
For b, I can show that $N(u) = p = x + xy + y^2$ by simply expanding out the norm of an arbitrary element of $R$. I also remember my professor mentioning that in a Euclidean domain, if the norm of an element is prime in the integers, then the element itself is prime in the ring. But how would I prove this?
Also, am I correct in saying that if the norm is a prime number, then the element itself cannot be a unit, as being a unit would imply that there exists $v$ in $R$ such that $uv = 1$. But $uv = 1$ implies $N(uv) = 1$, which is impossible since $N(uv) = N(u) N(v)$ for all $u,v$ in $R$, and $N(u) = p$, which implies $N(v) = 1/p $, and as $p >=2$, $N(v) < 1$, impossible.
Lastly, I would appreciate a hint for part c.
Edit : The ring in my question is different from the ring in the other one. Also, the questions themselves are different too. Please unmark it as a duplicate.