nLab archimedean valued field




Analytic geometry



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.


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} \,.


Archimedean valued fields include

Non-Archimedean valued fields include

Last revised on December 13, 2023 at 02:50:21. See the history of this page for a list of all contributions to it.