# 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 $\mid -\mid$ one defines

• its ring of integers to be

${k}^{\circ }:=\left\{x\in k\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}\mid x\mid \le 1\right\}\phantom{\rule{thinmathspace}{0ex}}.$k^\circ := \{x \in k \,|\, {\vert x\vert} \leq 1\} \,.

This is a local ring with maximal ideal

${k}^{\circ \circ }:=\left\{x\in k\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}\mid x\mid <1\right\}\phantom{\rule{thinmathspace}{0ex}}.$k^{\circ\circ} := \{x \in k \,|\, {\vert x\vert} \lt 1\} \,.
• The residue field of $k$ is the quotient

$\stackrel{˜}{k}:={k}^{\circ }/{k}^{\circ \circ }\phantom{\rule{thinmathspace}{0ex}}.$\tilde k := k^\circ / k^{\circ \circ} \,.

## Examples

Archimean fields include

Non-archimean fields include

Revised on January 9, 2012 06:08:33 by Toby Bartels (216.96.8.189)