Is $2 \in \mathbb{Z}[\sqrt{-2}]$ a unit, an irreducible element or a prime element?

Question

Is $2 \in \mathbb{Z}[\sqrt{-2}]$ a unit, an irreducible element or a prime element?

Or maybe is not anyone of these?

Answer 1

Since the norm is given by $N(a+b\sqrt{-2})=a^2+2b^2$ we have $N(2)=4$, which is different from $1$. Hence $2$ is not a unit. For the irreducible and prime elements in this ring see these duplicates:

Irreducible elements in $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?

Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$

The duplicates show that the only units are $\pm 1$, so that $\sqrt{-2}$ itself is not a unit. Now we have $$ 2=\sqrt{-2}\cdot (-\sqrt{-2}). $$