Definition:Unique Factorization Domain

Let $$\left({D, +, \circ}\right)$$ be an integral domain.

If, for all $$x \in D$$ such that $$x$$ is non-zero and not a unit of $$D$$:


 * 1) $$x$$ possesses a complete factorization in $$D$$;
 * 2) Any two complete factorizations of $$x$$ in $$D$$ are equivalent,

then $$D$$ is a unique factorization domain.