# Unique factorization domain

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A unique factorization domain is an integral domain in which an analog of the fundamental theorem of arithmetic holds. More precisely an integral domain $R$ is a unique factorization domain if for any element $r\in R$ which is not a unit:

• $r$ can be written in the form $r=p_1p_2\cdots p_n$ where $p_1,p_2,\ldots,p_n$ are (not necessarily distinct) irreducible elements in $R$.
• This representation is unique up to units and reordering, that is if $r = p_1p_2\cdots p_n = q_1q_2\cdots q_m$ where $p_1,p_2,\ldots,p_n$ and $q_1,q_2\ldots,q_m$ are all irreducibles then $m=n$ and there is some permutation $\sigma$ of $\{1,2\ldots,n\}$ such that for each $k$ there is a unit $u_k$ such that $p_k = u_kq_{\sigma(k)}$.

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