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'equivalent valuations'
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| Title of object: |
equivalent valuations |
| Canonical Name: |
EquivalentValuations |
| Type: |
Definition |
| Created on: |
2004-06-18 11:15:16 |
| Modified on: |
2004-06-18 11:30:08 |
| Classification: |
msc:13A18 |
Preamble:
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Content:
Let $K$ be a field. \, The equivalence of two 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 $|a|_2 < 1$ for all $a$ in $K.$
\end{enumerate}
It it easy to see that these conditions imply symmetry for both valuations (use $\frac{1}{a}$). \, Also, we have always
$$|a|_1 \leq 1 \, \equiv \, |a|_2 \leq 1.$$
Theorem: \, If the valuations $|\cdot|_1$ and $|\cdot|_2$ of $K$ are equivalent, then one is a positive power of the other,
$$|a|_1 = |a|_2^c \, \, \, \forall a \in K,$$
where $c$ is a positive constant. |
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