I was asked to provide an example of an integral domain in which the uniqueness provision of "unique factorization" fails. So my example is $\mathbb{Z}[\sqrt{-5}]$. My reasoning is that $6$ can be factored as $2*3$ and $(1 + \sqrt{-5})(1 + \sqrt{5})$, but I am having trouble to prove these are two different factorization. I took norm of each factor and say that
$$N(1 - \sqrt{-5}) = N(1 + \sqrt{-5}) = 6$$ while $$N(2) = 4, N(3) = 9$$. Since the norms of $2$ and $3$ do not divide the ones coming from the other factorizations, these are two different factorizations. Am I right? Thanks!