equivalent valuations

Let $K$ be a field. The equivalence of valuations $|\cdot|_{1}$ and $|\cdot|_{2}$ of $K$ may be defined so that

1.

$|\cdot|_{1}$ is not the trivial valuation;
2.

if $|a|_{1}<1$ then $|a|_{2}<1\qquad\forall a\in K.$

It it easy to see that these conditions imply 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 of the more general Krull valuations is defined to that they have common valuation rings.) Further, both valuations determine a common metric on $K$ .

Theorem.

Two valuations (of rank (http://planetmath.org/KrullValuation) one) $|\cdot|_{1}$ and $|\cdot|_{2}$ of $K$ are 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 .

Title	equivalent valuations
Canonical name	EquivalentValuations
Date of creation	2013-03-22 14:25:27
Last modified on	2013-03-22 14:25:27
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	18
Author	pahio (2872)
Entry type	Definition
Classification	msc 13A18
Related topic	DiscreteValuation
Related topic	IndependenceOfTheValuations
Defines	equivalence of valuations