Krull valuationKrullValuation2004-12-27 14:20:412007-04-05 06:33:12Definitionvaluationvalue grouprank of Krull valuationrank of valuation% this is the default PlanetMath preamble. as your knowledge
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% define commands here\textbf{Definition.}\, The mapping\, $|.|\!:\, K\to G$,\, where $K$ is a field and $G$ an ordered group equipped with zero, is a {\em Krull valuation} of $K$, if it has the properties
\begin{enumerate}
\item $|x| = 0 \,\,\Leftrightarrow\,\, x = 0$;
\item $|xy| = |x|\cdot|y|$;
\item $|x+y| \leqq \max\{|x|,\,|y|\}$.
\end{enumerate}
Thus the Krull valuation is more general than the usual \PMlinkname{valuation}{Valuation}, which is also characterized as \PMlinkescapetext{{\em valuation of rank 1}} and which has real values.\, The image\, $|K\!\smallsetminus\!\{0\}|$\, is called the {\em value group} of the Krull valuation; it is abelian.\, In general, the {\em rank of Krull valuation} \PMlinkescapetext{means} the \PMlinkname{rank}{IsolatedSubgroup} of the value group.
We may say that a Krull valuation is \PMlinkname{non-archimedean}{Valuation}.
\subsection*{Some values}
\begin{itemize}
\item $|1| = 1$\,\, because the Krull valuation is a group homomorphism from the multiplicative group of $K$ to the ordered group.
\item $|-1| = 1$\,\, because\, $1 = |(-1)^2| = |-1|^2$\,\, and 1 is the only element of the ordered group being its own inverse ($S\cap S^{-1} = \varnothing$).
\item $|-x| = |(-1)x| = |-1|\cdot|x| = |x|$
\end{itemize}
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\bibitem{Artin} {\sc Emil Artin}: {\em Theory of Algebraic Numbers}.\, Lecture notes.\, Mathematisches Institut, G\"ottingen (1959).
\end{thebibliography}