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Abstract Algebra

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Quadratic Ring : is $2$ Prime or just Irreducible? Is it neither?

ABSALG-AQN29A

Let $\mathbb{Z}$ be the ring of integers.

For a complex number $\alpha$ that satisfies a polynomial of degree $2$ in $\mathbb{Z}[x]$, let $R_\alpha$ be the ring:

$$\mathbb{Z}[\alpha]:=\mathbb{Z}+\mathbb{Z}\alpha=\{m+n\alpha\mid m, n\in\mathbb{Z}\}.$$

1) In the ring $\mathbb{Z}$, the integer $2$ is

.

2) In the ring $\mathbb{Z}[\sqrt{-1}]$, the integer $2$ is

.

3) In the ring $\mathbb{Z}\left[\cfrac{-1+\sqrt{-3}}{2}\right]$, the integer $2$ is

.
(You may use the fact that this ring is a Unique Factorization Domain (UFD).)

4) In the ring $\mathbb{Z}[\sqrt{-5}]$, the integer $2$ is

.