I want to show that the decomposition into irreducible factors in the ring
$$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$
is not unique, except for the order of factors and for elements being associated with each other.
Now I know that $6 = 2 · 3 = (1 + \sqrt{-5})(1-\sqrt{-5})$ and these are two different decompositions indeed. But how can it be shown that all of these factors are irreducible?
It might also be useful to know what the units of this ring are. Surely 1, -1 are units. But are there other ones, or are these the only ones?
Thanks in advance.