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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 $F$).

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

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