equivalent valuationsEquivalentValuations2004-06-18 11:15:162005-03-27 06:56:01Definitionvaluationequivalence of valuations% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}Let $K$ be a field. \,The {\em equivalence of valuations} $|\cdot|_1$ and $|\cdot|_2$ of $K$ may be defined so that
\begin{enumerate}
\item $|\cdot|_1$ is not the trivial valuation;
\item if \, $|a|_1 < 1$ then $|a|_2 < 1 \qquad \forall a \in K.$
\end{enumerate}
It it easy to see that these conditions imply \PMlinkescapetext{symmetry} for both valuations (use $\frac{1}{a}$). \,Also, we have always
$$|a|_1 \leqq 1 \, \Leftrightarrow \, |a|_2 \leqq 1;$$
so both valuations have a common valuation ring in the case they are non-archimedean. \,(The \PMlinkescapetext{equivalence} of the more general Krull valuations is defined to \PMlinkescapetext{mean} that they have common valuation rings.) \,Further, both valuations determine a common metric on $K$.
\begin{thmplain}
\,Two valuations (of \PMlinkname{rank}{KrullValuation} one) \,$|\cdot|_1$\, and \,$|\cdot|_2$\, of $K$ are \PMlinkescapetext{equivalent} iff one of them is a positive power of the other,
$$|a|_1 = |a|_2^c \qquad \forall a \in K,$$
where $c$ is a positive \PMlinkescapetext{constant}.
\end{thmplain}