Unique factorization theorem

In a UFD, every nonzero nonunit factors uniquely into irreducibles up to associates and order.

Unique factorization theorem

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.

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