# Contents

## Idea

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

So an archimedan 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

Archimean fields include

Non-archimean fields include

Revised on July 8, 2014 00:07:18 by Urs Schreiber (192.76.8.26)