nLab
archimedean field

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

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

This is a local ring with maximal ideal

$$k^{\circ \circ }:=\{x\in k\mid \mid x\mid <1\}.$$k^{\circ\circ} := \{x \in k \,|\, {\vert x\vert} \lt 1\} \,.

• The residue field of $k$ is the quotient

  $$\tilde{k}:=k^{\circ }/k^{\circ \circ }.$$\tilde k := k^\circ / k^{\circ \circ} \,.

Examples

Archimean fields include

• real numbers

Non-archimean fields include

• p-adic numbers.

Related concepts

• archimedean valued field