check whether $5$ is irreducible in $\Bbb{Z}[\sqrt{-2}]$

Question

As one can show that the integer $5$ is reducible in $\Bbb{Z}[i]$, as $5=(2-i)(2+i)$, how can I check whether $5$ is irreducible in $\Bbb{Z}[\sqrt{-2}]$?

Answer 1

Note that \begin{align*} \dfrac{{\bf{Z}}[X]}{\left<5,X^{2}+2\right>}\cong\dfrac{{\bf{Z}}_{5}[X]}{\left<X^{2}+2\right>} \end{align*} and $X^{2}+2$ has no root in ${\bf{Z}}_{5}[X]$.