 Failure of unique factorization in the ring of integers generated by square root of minus five

This integral domain embeds in $\mathbb{C}$ but has the inequivalent factorizations $6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$. The multiplicative norm $N(a+b\sqrt{-5})=a^2+5b^2$ takes neither the value two nor three, proving that the four displayed factors are irreducible elements. Its only units are $\pm 1$, so those two factorizations cannot be related by units and reordering.