In \(\mathbb{Z}[\sqrt{-5}]\), the element 6 has the factorizations \(2\cdot3\) and \((1+\sqrt{-5})(1-\sqrt{-5})\). Since the four factors are irreducible and not associates of each other, what follows?
There are two essentially different factorizations into irreducibles, contradicting uniqueness. The factors are irreducible because their norms 4, 9, 6 and 6 do not split into norms 2 or 3, which do not exist.
An integral domain \(D\) in which every nonzero nonunit element \(a\) is a product of irreducibles \(p_i\), and this factorization is unique up to order and associates.
\[a=p_1p_2\cdots p_r\]
More practice problems: unique factorization domain