22 lines
883 B
TeX
22 lines
883 B
TeX
\section{Euclidean Domains}
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\begin{definition}[Euclidean Domain (ED)]
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An integral domain $D$ is called a \textit{Euclidean domain} if there is a function $d$ (called the \textit{measure}) from nonzero elements of $D$ to the nonnegative integers such that
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\begin{enumerate}
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\item $d(a) \leq d(ab)$ for all nonzero $a,b \in D$; and
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\item if $a,b \in D,\ b \neq 0$, then there exist elements $q$ and $r$ in $D$ such that $a = bq + r$, where $r = 0$ or $d(r) < d(b)$.
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\end{enumerate}
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\end{definition}
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\begin{theorem}[ED Implies PID]
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Every Euclidean domain is a principal ideal domain.
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\end{theorem}
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\begin{corollary}[ED Implies UFD]
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Every Euclidean domain is a unique factorization domain.
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\end{corollary}
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\begin{theorem}[$\mathbf{D}$ a UFD Implies $\mathbf{D[x]}$ a UFD]
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If $D$ is a unique factorization domain, then $D[x]$ is a unique factorization domain.
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\end{theorem}
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