TITLE: Lack of unique factorization of ideals
QUESTION [13 upvotes]: I'm aware of the result that integral domains admit unique factorization of ideals iff they are Dedekind domains.
It's clear that $\mathbb{Z}[\sqrt{-3}]$ is not a Dedekind domain, as it is not integrally closed.  I'm having difficulty demonstrating directly that $\mathbb{Z}[\sqrt{-3}]$ does not admit unique factorization of ideals.
Obviously we only need one example of an ideal that doesn't have a unique factorization into prime ideals, but a good answer would provide either a process or some intuition in finding such an example.

REPLY [5 votes]: The arithmetic of nonmaximal quadratic orders comes up naturally when studying binary quadratic forms, so in particular Cox's book contains some discussion of this.
As it happens, when I taught a course based (sometimes closely, sometimes not) on Cox's book, I included a "case study" of factorization in the ring $\mathbb{Z}[\sqrt{-3}]$: see Section 3 of these notes.