= Failure of unique factorization in the ring of integers generated by square root of minus five
{title2=$\mathbb Z[\sqrt{-5}]$}
This <integral domain> embeds in $\mathbb C$ but has the inequivalent factorizations $6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})$. 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 $\pm1$, so those two factorizations cannot be related by units and reordering.
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