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Definition (Unique Factorization Domain)
Let $D$ be an integral domain. We say that $D$ is a unique factorization domain if the following two properties hold.
- If $d\in D$ is not a unit, and not zero, then we can write $d$ as a product of irreducibles over $D$.
- If we have written $d$ as a product of irreducibles over $D$ in two ways, say $d=p_1p_2\cdots p_n$ and $d=q_1q_2\cdots q_m$, then $n=m$ and after rearranging we have for each $i$ that $p_i=u_iq_i$ for some unit $u_i$.
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