value group of completion

Let k be a field and  ||  its non-archimedean valuation of rank one (  Then its value group|k{0}|  may be considered to be a subgroupMathworldPlanetmathPlanetmath of the multiplicative groupMathworldPlanetmath of .  In the completion K of the valued field k, the extensionPlanetmathPlanetmath of the valuationMathworldPlanetmath is defined by


when the Cauchy sequenceMathworldPlanetmathPlanetmathx1,x2,,xn,  of elements of k determines the element x of K.


The non-archimedean field k and its completion K have the same value group.

Proof.  Of course,  |k||K|.  Let  x=limnxn  be any non-zero element of K, where xj’s form a Cauchy sequence in k.  Then there exists a positive number n0 such that


for all  n>n0.  For all these values of n we have


according to the ultrametric triangle inequality.  Thus we see that  |K||k|.

Title value group of completion
Canonical name ValueGroupOfCompletion
Date of creation 2013-03-22 14:58:14
Last modified on 2013-03-22 14:58:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 13F30
Classification msc 13J10
Classification msc 13A18
Classification msc 12J20
Related topic KrullValuation
Related topic ExtensionOfValuationFromCompleteBaseField
Defines value group of the completion