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 .