= Unique factorization in an integral domain
{title2=$a=u\prod_{i=1}^r p_i$}
= Unique factorization
{synonym}
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>.
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