# Math Help - Euclidean Domain

1. ## Euclidean Domain

Suppose $\rho \in \mathbb{Z}[i]$ and that $\lambda(\rho) = r$ is prime in $\mathbb{Z}$. Show that $\rho$ is prime in $\mathbb{Z}[i]$.

I think the easiest way is to use the contrapositive, but I'm not sure.

2. Originally Posted by chiph588@
Suppose $\rho \in \mathbb{Z}[i]$ and that $\lambda(\rho) = r$ is prime in $\mathbb{Z}$. Show that $\rho$ is prime in $\mathbb{Z}[i]$.

I think the easiest way is to use the contrapositive, but I'm not sure.
If $\rho = \rho_1 \rho_2$ for non-units $\rho_1,\rho_2$ then $r=\lambda (\rho) = \lambda (\rho_1)\lambda (\rho_2)$. This is a contradiction. Can you see why?