\section{Unique Factorization Domains}

\begin{definition}
	An integral domain $D$ is a \textit{unique factorization domain} if
	\begin{enumerate}
		\item every nonzero element of $D$ that is not a unit can be written as a product of irreducibles of $D$; and
		\item the factorization into irreducibles is unique up to associates and the order in which the factors appear.
	\end{enumerate}
\end{definition}

\begin{lemma}[Ascending Chain Condition for a PID]
	In a principal ideal domain, any stricly increasing chain of ideals $I_1 \subset I_2 \subset \dots$ must be finite in length.
\end{lemma}

\begin{theorem}[PID Implies UFD]
	Every principal ideal domain is a unique factorization domain.
\end{theorem}

\begin{corollary}[$\mathbf{\F[x]}$ Is a UFD]
	Let $\F$ be a field. Then $\F[x]$ is a unique factorization domain.
\end{corollary}