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'Euclidean valuation'
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| Title of object: |
Euclidean valuation |
| Canonical Name: |
EuclideanValuation |
| Type: |
Definition |
| Created on: |
2002-05-27 23:20:49 |
| Modified on: |
2008-02-10 14:51:59 |
| Classification: |
msc:13F07 |
| Synonyms: |
Euclidean valuation=Euclidean norm |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
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Content:
Let $D$ be an integral domain. A \emph{Euclidean valuation} is a function from the nonzero elements of $D$ to the nonnegative integers $\nu \colon D \setminus \{0_D\} \to \{ x \in \mathbb{Z} : x \ge 0 \}$ such that the following hold:
\begin{itemize}
\item For any $a,b\in D$ with $b\neq 0_D$, there exist $q,r\in D$ such that $a=bq+r$ with $\nu(r)<\nu(b)$ or $r=0_D$.
\item For any $a,b\in D$ both non-zero, $\nu(a)\leq\nu(ab)$.
\end{itemize}
Euclidean valuations are important because they let us define greatest common divisors and use Euclid's algorithm. Some facts about Euclidean valuations include:
\begin{itemize}
\item The \PMlinkname{minimal}{MinimalElement} value of $\nu$ is $\nu(1_D)$. That is, $\nu(1_D)\leq\nu(a)$ for any nonzero element $a$ of $D$.
\item $u\in D$ is a unit if and only if $\nu(u)=\nu(1_D)$.
\end{itemize}
Note that an integral domain is a Euclidean domain if and only if it has a Euclidean valuation.
Below are some examples of Euclidean domains and their Euclidean valuations:
\begin{itemize}
\item $\mathbb{Z}$ is a Euclidean domain, and absolute value is its Euclidean valuation.
\item If $F$ is a field, then $F[x]$, the ring of polynomials over $F$, is a Euclidean domain with the Euclidean valuation of degree: If $n$ is a nonnegative integer and $a_0,\dots,a_n\in F$ with $a_n\neq 0_F$, then
\[
\nu\left(\sum_{j=0}^n a_jx^j\right)=n.
\]
\end{itemize} |
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