22 lines
841 B
TeX
22 lines
841 B
TeX
\section{Unique Factorization Domains}
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\begin{definition}
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An integral domain $D$ is a \textit{unique factorization domain} if
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\begin{enumerate}
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\item every nonzero element of $D$ that is not a unit can be written as a product of irreducibles of $D$; and
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\item the factorization into irreducibles is unique up to associates and the order in which the factors appear.
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\end{enumerate}
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\end{definition}
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\begin{lemma}[Ascending Chain Condition for a PID]
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In a principal ideal domain, any stricly increasing chain of ideals $I_1 \subset I_2 \subset \dots$ must be finite in length.
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\end{lemma}
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\begin{theorem}[PID Implies UFD]
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Every principal ideal domain is a unique factorization domain.
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\end{theorem}
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\begin{corollary}[$\mathbf{\F[x]}$ Is a UFD]
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Let $\F$ be a field. Then $\F[x]$ is a unique factorization domain.
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\end{corollary}
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