I need some help with the following: I have already proved that $\mathbb Z[i]/(p) \cong \mathbb F_p[x]/(x^2+1)$ however, I cannot figure out how to prove that:
- (p) is prime in $\mathbb Z[i]$ if and only if there is no such
$\mathbb a \in \mathbb F_p$ such that $\mathbb a^2+1=0 $
- It happens if and only if $\mathbb p\equiv 3\pmod 4$
I managed to do one direction of the first of these. Since if there was such an $\mathbb a$ one could consider $\mathbb (x-a)(x+a)$ which gives $\mathbb 0$ as a result in $\mathbb F_p[x]/(x^2+1)$ while both of the factors are nonzero thus $\mathbb F_p[x]/(x^2+1)$ is not an integral domain, which means that $\mathbb Z[i]/(p)$ is not an integral domain either, which in turn gives us that $\mathbb (p)$ is not a prime ideal of $\mathbb Z[i]$.
However, I am stuck with the other direction and the second part. Could you help me out? Thanks!