# Thread: 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 $\displaystyle p \equiv 1 (mod \ 4)$, 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.

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