Primes in $\mathbb{Z}[i]$ I need a bit of help with this problem
Let $x \in \mathbb{Z}[i]$ and suppose $x$ is prime, therefore $x$ is not a unit and cannot be written as a product of elements of smaller norm. Prove that $N(x)$ is either prime in $\mathbb{Z}$ or else $N(x) = p^2$ for some prime $p \in \mathbb{Z}$.
thanks.

Hint $\ $ Prime $\rm\:w\mid ww' = p_1^{k_1}\!\cdots p_n^{k_n}\:\Rightarrow\:w\mid p_i\:\Rightarrow\:w'\mid p_i' = p_i\:\Rightarrow\:N(w) = ww'\mid p_i^2$
Here $'$ denotes the complex conjugation.
