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equivalent valuations
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(Definition)
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Let $K$ be a field. The equivalence of valuations $|\cdot|_1$ and $|\cdot|_2$ of $K$ may be defined so that
- $|\cdot|_1$ is not the trivial valuation;
- if $|a|_1 < 1$ then $|a|_2 < 1 \qquad \forall a \in K.$
It it easy to see that these conditions imply 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 equivalence of the more general Krull valuations is defined to mean that they have common
valuation rings.) Further, both valuations determine a common metric on $K$ .
Theorem 1 Two valuations (of rank one) $|\cdot|_1$ and $|\cdot|_2$ of $K$ are 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 constant.
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"equivalent valuations" is owned by pahio.
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Cross-references: power, positive, iff, metric, Krull valuations, non-archimedean, valuation ring, valuations, imply, easy to see, trivial valuation, field
There are 4 references to this entry.
This is version 15 of equivalent valuations, born on 2004-06-18, modified 2005-03-27.
Object id is 5932, canonical name is EquivalentValuations.
Accessed 3751 times total.
Classification:
| AMS MSC: | 13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations) |
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Pending Errata and Addenda
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