# nLab archimedean valued field

Contents

### 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 valued 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 valued, in which case $k$ is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.

### Non-archimedean valued fields

For $k$ a non-archimedean valued field for some non-archimedean absolute value ${\vert -\vert}$ one defines

• its ring of integers to be

$k^\circ := \{x \in k \,|\, {\vert x\vert} \leq 1\} \,.$

This is a local ring with maximal ideal

$k^{\circ\circ} := \{x \in k \,|\, {\vert x\vert} \lt 1\} \,.$
• The residue field of $k$ is the quotient

$\tilde k := k^\circ / k^{\circ \circ} \,.$

## Examples

Archimedean valued fields include

Non-archimedean valued fields include

Last revised on May 28, 2021 at 15:42:26. See the history of this page for a list of all contributions to it.