complete ultrametric field

A field K equipped with a non-archimedean valuation||  is called a non-archimedean field or also an ultrametric field, since the valuationMathworldPlanetmathPlanetmath the ultrametricd(x,y):=|x-y|  of K.


Let (K,d) be a completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( ultrametric field.  A necessary and sufficient condition for the convergence of the series

a1+a2+a3+ (1)

in K is that

limnan= 0. (2)

Proof.  Let ε be any positive number.  When (1) convergesPlanetmathPlanetmath, it satisfies the Cauchy condition and therefore exists a number mε such that surely


for all  mmε;  thus (2) is necessary.  On the contrary, suppose the validity of (2).  Now one may determine such a great number nε that

|am|<ε  mnε.

No matter how great is the natural numberMathworldPlanetmath n, the ultrametric then guarantees the inequality


always when  mnε.  Thus the partial sums of (1) form a Cauchy sequencePlanetmathPlanetmath, which converges in the complete field.  Hence the series (1) converges, and (2) is sufficient.

Title complete ultrametric field
Canonical name CompleteUltrametricField
Date of creation 2013-03-22 14:55:37
Last modified on 2013-03-22 14:55:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Theorem
Classification msc 12J10
Classification msc 54E35
Related topic Series
Related topic NecessaryConditionOfConvergence
Related topic ExtensionOfValuationFromCompleteBaseField
Related topic PropertiesOfNonArchimedeanValuations
Defines ultrametric field
Defines non-archimedean field