nLab
archimedean field

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 kk a non-archimedean field for some non-archimedean absolute value ||{\vert -\vert} one defines

  • its ring of integers to be

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

This is a local ring with maximal ideal

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

    k˜:=k /k . \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)