Let p be prime with $\displaystyle p \equiv 1 (mod \ 4) $, show that p is not irreducible in Z[i].

Proof. Now from the previous problem, I know that p \ n^2 + 1 = (n+i)(n-i).

If p is irreducible, then p is a prime and it divides either (n+i) or (n-i). I'm trying to show that is impossible, any hint?

Thank you.