Can anyone give me a hint with proving the following,
Let $\alpha\in\mathbb Z[i],$ and let $P$ be a non-zero prime ideal of $\mathbb Z[\alpha].$ Show that the quotient $\mathbb Z[\alpha]/P$ is a finite ring.
My ideas have been to try use the euclidean function for Z[i] to show all coset representatives have norm bounded but I haven't been able to get this to work.