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equivalent valuations
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(Definition)
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Let be a field. The equivalence of valuations and of may be defined so that
is not the trivial valuation;
- if
then

It it easy to see that these conditions imply symmetry for both valuations (use
). Also, we have always
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 .
Theorem 1 Two valuations (of rank one)  and  of  are equivalent iff one of them is a positive power of the other,
where  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 3136 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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