$S^{-1}(\mathbb{Z}[i])$, where $S=\{x\in \mathbb{Z}|5\nmid x\}$.

Question

Let $S=\{x\in \mathbb{Z}|5\nmid x\}$. I would like to know all the prime ideals of $S^{-1}(\mathbb{Z}[i])$.

My attempt: Since $S\subset \mathbb{Z}$, the given question can be rewritten as $(\mathbb{Z}[x]/(x^2+1))_{(5)}=\mathbb{Z}_{(5)}[x]/(x^2+1)=\mathbb{Z}_{(5)}[i]$. How do we determine how many prime ideals are there, and what they are?

Answer 1

In fact $S=\mathbb Z-5\mathbb Z$ and the prime ideals of $S^{-1}(\mathbb Z[i])$ are of the form $S^{-1}P$ with $P$ prime in $\mathbb Z[i]$ and $P\cap S=\emptyset$, that is, $P\cap \mathbb Z=(0)$ or $5\mathbb Z$. If $P=(a+bi)$, from $P\cap\mathbb Z=(0)$ we get $P=\dots$, and from $P\cap\mathbb Z=5\mathbb Z$ we get $P=\dots$.