nLab archimedean valued field

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

Non-Archimedean valued fields

For $k$ a non-Archimedean valued 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

Archimedean valued fields include

• real numbers
• complex numbers

Non-Archimedean valued fields include

• p-adic numbers.

Related concepts

• archimedean field

• absolute value

• non-Archimedean analytic geometry