nLab
archimedean valued field

Contents

Context

Algebra

Analytic geometry

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 kk a field complete with respect to an absolute value ||{\vert - \vert} either the absolute value is archimedean valued, in which case kk is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.

Non-archimedean valued fields

For kk a non-archimedean valued 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

Archimedean valued fields include

Non-archimedean valued fields include

Last revised on May 28, 2021 at 11:42:26. See the history of this page for a list of all contributions to it.