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# 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 symmetry 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 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 $K$.

Defines:

equivalence of valuations

Related:

DiscreteValuation, IndependenceOfTheValuations

Type of Math Object:

Definition

Major Section:

Reference

Parent:

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## Mathematics Subject Classification

13A18*no label found*

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