A nice discussion, but a bit of a nitpick: in Z[sqrt(-5)], 2, 3, 1+sqrt(-5), and 1-sqrt(-5) are actually irreducible, not prime.

In an integral domain D, a nonzero element x is called irreducible if x is not a unit and whenever x = ab (for a, b in D), then one of a or b is a unit.

On the other hand, a nonunit element x in D is called prime if for all a, b in D, if x divides ab then x divides a or x divides b (by "x divides ab", I mean that there's some element--call it y--in D such that xy = ab).

In Z (or any unique factorization domain[1]), these concepts coincide. In Z[sqrt(-5)], however, there are irreducible elements that are not prime. In particular, 2 is irreducible in Z[sqrt(-5)], but it isn't prime, since 2 divides (1+sqrt(-5))*(1-sqrt(-5)), but 2 divides neither 1+sqrt(-5) nor 1-sqrt(-5).

[1]: https://en.wikipedia.org/wiki/Unique_factorization_domain