Element of Principal Ideal Domain is Finite Product of Irreducible Elements
Let $R$ be a principal ideal domain.
Let $p \in R$ such that $p \ne 0$ and $p$ is not a unit.
Then there exist irreducible elements $p_1, \ldots, p_n$ such that $p = p_1 \cdots p_n$.
If $p$ is irreducible, it is proven.
Suppose $p$ is not irreducible.
Then $p = r_1 r_2$ where neither $r_1$ nor $r_2$ are units.
If $r_1$ and $r_2$ are irreducible, then the proof is complete.
If this process finishes in a finite number of steps, the proof is complete.
From Principal Ideal Domain fulfills Ascending Chain Condition, this cannot happen.
Thus, the process ends in a finite number of steps.