Definition: Absolute Values and Non-Archimedean Absolute Values of Fields

Let $(F,\oplus,\odot)$ be a field. An absolute value $|\cdot|:F\to \mathbb R$ is a real-valued function fulfilling the following properties:

  1. $|x|\ge 0$ for all $x\in F.$
  2. $|x|=0$ if and only if $x=0\in F.$
  3. $|x\odot y|=|x|\cdot |y|$ for all $x,y\in F.$
  4. $|x\oplus y|\le |x|+|y|$ for all $x,y\in F$ triangle inequality).

An absolute value, in which the 4th axiom is replaced by

4. $|x\oplus y|\le \max(|x|,|y|)$

(i.e. the maximum of the absolute values), is called non-archimedean (or a valuation of $F$).


Definitions: 1 2
Proofs: 3 4 5
Propositions: 6 7 8

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  1. Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition