Let $R=\mathbb Z[e^{i\pi/3}]=\{a+be^{i\pi/3}\mid a,b\in \mathbb Z\}\subseteq \mathbb C$
Prove that if $q$ is a prime element in $R$, then either $q$ ~ $u$ for some $u \in R$ such that $N(u)=p$, where $p \in \mathbb{Z}$ is prime, or $q$ ~ $p$ for some prime number $p\in \mathbb{Z}$ such that $p\neq N(u)$ for all $u\in R$.
(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 $R$.
My attempt : I can't figure out how to show that long division works in $R$, but other than that, I've done parts a till c. I think I have to use parts b and c for the question. b and c involved me 'converting' the problem from one in $R$ to one in $N$ by using the norm. But I really have no idea how to begin. It's an 'or' statement, so I have to assume that $q$ is prime and one condition doesn't hold, then the other one must hold. But I don't know how to actually show this.
