# nLab archimedean valued field

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Analytic geometry

analytic geometry (complex, rigid, global)

## Basic concepts

analytic function

analytification

GAGA

# Contents

## Definition

A (complete) archimedean valued field is a field equipped with an archimedean absolute value (and complete with respect to it).

A non-archimedean field is one that is not, hence one whose norm satisfies the ultrametric triangle inequality.

## Properties

One of Ostrowski's theorems says that for $k$ a field complete with respect to an absolute value ${\vert - \vert}$ either the absolute value is archimedean, in which case $k$ is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.

Revised on July 17, 2014 14:55:01 by Urs Schreiber (82.136.246.44)