# Contents

## Idea

An archimedean field is an ordered field in which every element is bounded above by a natural number.

So an archimedean field has no infinite elements (and thus no non-zero infinitesimal elements).

## Non-archimedean fields

For $k$ a non-archimedean 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 fields include

Non-archimedean fields include

Last revised on July 7, 2017 at 11:59:19. See the history of this page for a list of all contributions to it.