I'm struggling with the following problem.
Given the integral domain $\mathbb{Z}[\sqrt{5}]:=\{a+b\sqrt{5}\,/\,a,b\in\mathbb{Z}\}$, we define the norm $N:\mathbb{Z}[\sqrt{5}]\to\mathbb{Z}\;/\;N(a+b\sqrt{5})=a^2-5b^2$. I have succesfully proven that $N$ is multiplicative, i.e., $N(xy)=N(x)N(y)$; and that $x$ is a unit of $\mathbb{Z}[\sqrt{5}]$ if and only if $N(x)$ is a unit of $\mathbb{Z}$. Next, the problem asks me to prove the following statement:
"If $N(x)$ is prime in $\mathbb{Z}$, then $x$ is prime in $\mathbb{Z}[\sqrt{5}].$"
This is proving to be quite the challenge. I understand that an element $x$ is prime if $x|mn\Rightarrow x|m \lor x|n$. So far, I've only been able to prove that if $N(x)$ is prime, $x$ is irreducible.
After some time, I began to think that maybe the source of the problem was mistaken and meant irreducible instead of prime, seeing as both definitions are equivalent in $\mathbb{Z}$. However, I haven't been able to find a suitable counterexample so far. So, my question is, is the claim true? If so, how could I prove it?