The prime element in Eisenstein ring

Question

I have to proof that prime $p\in \mathbb {Z}$ is prime in Eisenstein ring $\mathbb{Z}[\omega]$ if and only if $\mathbb{F}_p[x]/(x^2+x+1)$ is a field. I know that there is norm $N$ on $\mathbb{Z}[\omega]$ such that $N(a+b\omega)=a^2-ab+b^2$. Also I proved Eisenstein ring is Euclidean ring. Can anyone help me? Thanks a lot.

Answer 1

This is almost immidiate given that $\omega^2+\omega+1=0$, which gives $$\Bbb Z[\omega]\cong\Bbb Z[x]/(x^2+x+1)\tag{1}$$

• $\Bbb Z[\omega]$ is a UFD, so any prime $\pi\in\Bbb Z[\omega]$ is also irreducible.

• $\Bbb Z[\omega]$ is a PID, meaning that $\Bbb Z[\omega]/(\pi)$ is a field for any irreducible $\pi\Bbb Z[\omega]$.

All in all,

$$p\in\Bbb Z\text{ is prime in }\Bbb Z[\omega]\iff\Bbb Z[\omega]/(p)\text{ is a field}\\\Bbb Z[\omega]/(p)\text{ field}\iff\Bbb Z_p[x]/(x^2+x+1)\text{ is a field}$$