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 $$ then $$
It it easy to see that these conditions imply for both valuations^{} (use $\frac{1}{a}$). Also, we have always
$${a}_{1}\leqq 1\iff {a}_{2}\leqq 1;$$ 
so both valuations have a common valuation ring^{} in the case they are nonarchimedean. (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,
$${a}_{1}={a}_{2}^{c}\mathit{\hspace{1em}\hspace{1em}}\forall a\in K,$$ 
where $c$ is a positive .
Title  equivalent valuations 

Canonical name  EquivalentValuations 
Date of creation  20130322 14:25:27 
Last modified on  20130322 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 