Unique factorization in an integral domain ($a=u{\prod }_{i=1}^r{p}_i$)

Every nonzero nonunit in a unique factorization domain is a finite product of irreducible elements. Such a factorization is unique up to permutation and multiplication of the factors by units. Irreducibles are prime elements, so divisibility can be checked by comparing their multiplicities. This applies to a polynomial ring over a field and is useful for intersections of localizations inside its fraction field.