Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$ Let $w\in\mathbb C$ be such that $w^3=1$ and $w\neq1$. Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$, and describe $\mathbb Z[w]/(2)$.
What I wanted to do is to show that $\mathbb Z[w]$ is a PID, then $(2)$ would be prime because it's principal. To show that, I thought it would be easier to show a stronger claim that $\mathbb Z[w]$ is an Euclidean domain, but wasn't able to.
My textbook doesn't do a terrific job of explaining the notation here so I'm half-guessing that $\mathbb Z[w]=\{a+bw+cw^2:a,b,c\in\mathbb Z\}$.
Then I tried to construct a Euclidean function. Since $(a+bi)\mapsto(a^2+b^2)$ works for $\mathbb Z[i]$, I tried the similar $(a+bw+cw^2)\mapsto(a^2+b^2+c^2)$, but didn't know how to prove any of the properties for it.

The ring $\mathbf Z[w]$ is isomorphic to the quotient ring $A=\mathbf Z[x]/(x^2+x+1)$. To show $2$ is prime, you have to show $A/2A$ is an integral domain.
We have:
$$A/2A \simeq \mathbf Z[x]/(2,x^2+x+1)\simeq \mathbf Z/2\mathbf Z[x]/(x^2+x+1)$$
so it is enough to show the polynomial $x^2+x+1$ is irreducible over $\mathbf Z/2\mathbf Z$, which means it has no root. It is easy to see the associated polynomial function is the constant $1$.
