Editing Absolute value (section)

===Fields===
{{Main|Absolute value (algebra)}}
The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function&nbsp;{{mvar|v}} on a [[field (mathematics)|field]]&nbsp;{{mvar|F}} is called an ''absolute value'' (also a ''modulus'', ''magnitude'', ''value'', or ''valuation'')<ref>Shechter, [https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 p.&nbsp;260]. This meaning of ''valuation'' is rare. Usually, a [[valuation (algebra)|valuation]] is the logarithm of the inverse of an absolute value</ref> if it satisfies the following four axioms:

:{| cellpadding=10
|-
|<math>v(a) \ge 0 </math>
|Non-negativity
|-
|<math>v(a) = 0 \iff a = \mathbf{0} </math>
|Positive-definiteness
|-
|<math>v(ab) = v(a) v(b) </math>
|Multiplicativity
|-
|<math>v(a+b)  \le v(a) + v(b)  </math>
|Subadditivity or the triangle inequality
|}

Where '''0''' denotes the [[additive identity]] of&nbsp;{{mvar|F}}. It follows from positive-definiteness and multiplicativity that {{math|1=''v''('''1''') = 1}}, where '''1''' denotes the [[multiplicative identity]] of&nbsp;{{mvar|F}}. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If {{mvar|v}} is an absolute value on&nbsp;{{mvar|F}}, then the function&nbsp;{{mvar|d}} on {{math|''F'' × ''F''}}, defined by {{math|1=''d''(''a'', ''b'') = ''v''(''a'' − ''b'')}}, is a metric and the following are equivalent:

* {{mvar|d}} satisfies the [[ultrametric]] inequality <math>d(x, y) \leq \max(d(x,z),d(y,z))</math> for all {{mvar|x}}, {{mvar|y}}, {{mvar|z}} in&nbsp;{{mvar|F}}.
* <math display="inline"> \left\{ v\left( \sum_{k=1}^n \mathbf{1}\right) : n \in \N \right\} </math> is [[bounded set|bounded]] in&nbsp;'''R'''.
* <math> v\left({\textstyle \sum_{k=1}^n } \mathbf{1}\right) \le 1\ </math> for every <math>n \in \N</math>.
* <math> v(a) \le 1 \Rightarrow v(1+a) \le 1\ </math> for all <math>a \in F</math>.
* <math> v(a + b) \le \max \{v(a), v(b)\}\ </math> for all <math>a, b \in F</math>.

An absolute value which satisfies any (hence all) of the above conditions is said to be '''non-Archimedean''', otherwise it is said to be [[Archimedean field|Archimedean]].<ref>Shechter, [https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 pp.&nbsp;260–261].</ref>