# equivalent valuations

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

1. 1.

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

2. 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 EquivalentValuations 2013-03-22 14:25:27 2013-03-22 14:25:27 pahio (2872) pahio (2872) 18 pahio (2872) Definition msc 13A18 DiscreteValuation IndependenceOfTheValuations equivalence of valuations