# Math Help - Show that a prime is not irreducible in Z[i]

1. ## Show that a prime is not irreducible in Z[i]

Let p be prime with $p \equiv 1 (mod \ 4)$, 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.

Let p be prime with $p \equiv 1 (mod \ 4)$, show that p is not irreducible in Z[i].
By your other problem if $p\equiv 1 (\bmod 4)$ then $p|(n^2+1)$ for some $n\in \mathbb{Z}$. Thus, $p|(n+i)(n-i)$, assume that $p$ is irreducible then $p|(n+i)$ or $p|(n-i)$. If $p|(n+i)$ then $n+i = p(a+bi)$ by definition for some $a,b\in \mathbb{Z}$. Thus, $n+i = pa+pbi \implies 1 = pb$ which is an impossibility. Similarly, $p$ cannot divide $n-i$. Thus, the assumption that $p$ is irreducible is false.