In $\mathbb Z[\sqrt{5}]$, $2$ and $1+\sqrt{5}$ are irreducible but not prime.
To show irreducible
I tried that there exists $\alpha$ and $\beta$ such that
$$2=\alpha\cdot \beta $$ $$N(2)=N(\alpha)\cdot N(\beta)$$ $$4=N(\alpha)\cdot N(\beta)$$
then there are 3 possibilities (namely $1.4 , 4.1 , 2.2$)
Now,I have to show that $N(\alpha)=2$ is not possible
that is $|a^2-5b^2|=2$
$a^2-5b^2=\pm 2$
How can I show that this is not possible? and how can i show that $2$ is not prime in $Z[\sqrt{5}]$?