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[parent] equivalent valuations (Definition)

Let $ K$ be a field. The equivalence of valuations $ \vert\cdot\vert _1$ and $ \vert\cdot\vert _2$ of $ K$ may be defined so that

  1. $ \vert\cdot\vert _1$ is not the trivial valuation;
  2. if $ \vert a\vert _1 < 1$ then $ \vert a\vert _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

$\displaystyle \vert a\vert _1 \leqq 1 \, \Leftrightarrow \, \vert a\vert _2 \leqq 1;$
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) $ \vert\cdot\vert _1$ and $ \vert\cdot\vert _2$ of $ K$ are equivalent iff one of them is a positive power of the other,
$\displaystyle \vert a\vert _1 = \vert a\vert _2^c \qquad \forall a \in K,$
where $ c$ is a positive constant.



"equivalent valuations" is owned by pahio.
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See Also: discrete valuation, independence of valuations

Also defines:  equivalence of valuations

This object's parent.

Attachments:
Ostrowski's valuation theorem (Theorem) by pahio
proof of theorem on equivalent valuations (Proof) by rspuzio
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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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