Definition:Complete Factorization

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

Let $$x$$ be a non-zero non-unit element of $$D$$.

A complete factorisation of $$x$$ in $$D$$ is a tidy factorization:

$$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_m$$

such that all of $$y_1, y_2, \ldots, y_n$$ are irreducible.