ultrametric triangle inequality

Theorem 1.

Let K be a field and G an ordered group equipped with zero.  Suppose that the function  ||:KG  satisfies the postulates 1 and 2 of Krull valuation.  Then the non-archimedean or ultrametric triangle inequality

3.     |x+y|max{|x|,|y|}

in the field is with the condition

(*) |x|1|x+1|1.

Proof.  The value  y=1  in the ultrametric triangle inequality gives the (*) as result.  Secondly, let’s assume the condition (*).  Let x and y be non-zero elements of the field K (if  xy=0  then 3 is at once verified), and let e.g.  |x||y|.  Then we get  |xy|=|x||y|-11,  and thus according to (*),


So we see that  |x+y||y|=max{|x|,|y|}.

Theorem 2.

The Krull valuation (and any non-archimedean valuation (http://planetmath.org/Valuation))  ||  of the field K satisfies the sharpening


of the ultrametric triangle inequality.

Proof.  Let e.g.  |x|>|y|.  Surely  |x+y||x|,  but also  |x|=|(x+y)-y|max{|x+y|,|y|};  this maximum is |x+y| since otherwise one would have  |x||y|.  Thus the result is:  |x+y|=|x|.

Note.  The metric defined by a non-archimedean valuation of the field K is the ultrametric of K.  Theorem 2 implies, that every triangle of K with vertices A, B, C (K) is isosceles:  if  |B-C||C-A|,  then  |A-B|=max{|B-C|,|C-A|}.

Theorem 3.

The valuation (http://planetmath.org/Valuation)  ||:K  of the field K is archimedeanPlanetmathPlanetmathPlanetmath if and only if the set


of the “values” of the multiplesMathworldPlanetmathPlanetmath of the unity is not boundedPlanetmathPlanetmathPlanetmathPlanetmath.

Proof.  If || is non-archimedean, then  |n1|=|1++1|max{|1|}=1,  and the multiples are bounded.  Conversely, let  |n1|<Mn+.  Now one obtains, when  |x|1:


or  |x+1|<(n+1)Mn   for all n.  As n tends to infinity, this nth root has the limit 1.  Therefore one gets the limit inequality|x+1|1,  i.e. the valuation is non-archimedean.


  • 1 Emil Artin: Theory of Algebraic NumbersMathworldPlanetmath.  Lecture notes.  Mathematisches Institut, Göttingen (1959).
Title ultrametric triangle inequality
Canonical name UltrametricTriangleInequality
Date of creation 2013-03-22 14:54:15
Last modified on 2013-03-22 14:54:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 25
Author pahio (2872)
Entry type Theorem
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic MaximalNumber
Related topic PAdicCanonicalForm
Related topic UltrametricSpace
Related topic MinimalAndMaximalNumber
Related topic ExponentValuation2
Defines non-archimedean triangle inequality