Unique factorization theorem - Imported Knowlpedia Content

Unique factorization theorem: Let $R$ be a unique factorization domain. Every nonzero element of $R$ that is not a unit can be written as a finite product of irreducible elements. Moreover, if

r = u\,p_1\cdots p_m = v\,q_1\cdots q_n

with $u,v$ units and $p_i,q_j$ irreducible, then $m=n$ and after reordering, each $p_i$ is associated to $q_i$ . In particular, irreducibles are prime in a UFD.

Remarks

This theorem is the foundational reason gcd/lcm notions behave well in UFDs and underlies factorization results in polynomial rings via Gauss-type arguments.