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Definition: Absolute Values and NonArchimedean Absolute Values of Fields
Let $(F,\oplus,\odot)$ be a field. An absolute value $\cdot:F\to \mathbb R$ is a realvalued function fulfilling the following properties:
 $x\ge 0$ for all $x\in F.$
 $x=0$ if and only if $x=0\in F.$
 $x\odot y=x\cdot y$ for all $x,y\in F.$
 $x\oplus y\le x+y$ 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 nonarchimedean (or a valuation of $F$).
Notes
 The absolute value can be defined for all fields, not only for ordered fields. This is because an absolute value is simply a function fulfilling some properties which do not require the domain $F$ to be an ordered field.
 In some special literature, you will find definitions of absolute values that require the field $F$ to be ordered. This is the case in definitions like this $$x:=\begin{cases}x&\text{if }x \ge 0,\\x&\text{else }x < 0.\end{cases}$$ You should be aware that this is only a special case definition applicable for only some common fields like the field of real numbers $(\mathbb R, +,\cdot).$ It would be not applicable for any unordered field. For instance, the unordered field of complex numbers $(\mathbb C, +,\cdot)$ requires another definition of an absolute value.
 The definition of an absolute value given here is applicable for all kinds of fields.
Mentioned in:
Definitions: 1 2
Proofs: 3 4 5
Propositions: 6 7 8
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References
Bibliography
 Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition