| Version 3 |
Version 2 |
| Let $D$ be an integral domain. An Euclidean valuation is a function from non-zero elements to the non-negative integers |
Let $D$ be an integral domain. An Euclidean valuation is a function from non-zero elements to the non-negative integers |
| $$\nu :D-\{0\}\to\Z^+\cup\{0\}$$ |
$$\nu :D-\{0\}\to\Z^+\cup\{0\}$$ |
| such that |
such that |
| \begin{itemize} |
\begin{itemize} |
| \item For any $a,b\in D$, $b\neq 0$, there exist $q,r\in D$ such that $a=bq+r$ with $\nu(r)<\nu(b)$ or $r=0$. |
\item For any $a,b\in D$, $b\neq 0$, there exist $q,r\in D$ such that $a=bq+r$ with $\nu(r)<\nu(b)$ or $r=0$. |
| \item For any $a,b\in D$ both non-zero, $\nu(a)\leq\nu(ab)$. |
\item For any $a,b\in D$ both non-zero, $\nu(a)\leq\nu(ab)$. |
| \end{itemize} |
\end{itemize} |
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| Euclidean valuations are important because they let us define greatest common divisors and use Euclid's algorithm. Some facts about Euclidean valuations: |
Euclidean valuations are important because they let us define greatest common divisors and use Euclid's algorithm. Some facts about Euclidean valuations: |
| \begin{itemize} |
\begin{itemize} |
| \item The value $\nu(1)$ is minimal. That is, $\nu(1)\leq\nu(a)$ for any nonzero element of $D$. |
\item The value $\nu(1)$ is minimal. That is, $\nu(1)\leq\nu(a)$ for any nonzero element of $D$. |
| \item $u\in D$ is an unit if an only if $\nu(u)=\nu(1)$. |
\item $u\in D$ is an unit if an only if $\nu(u)=\nu(1)$. |
| \end{itemize} |
\end{itemize} |
