value group of completion
Let k be a field and |⋅| its non-archimedean valuation of rank one (http://planetmath.org/KrullValuation). Then its value group |k∖{0}| may be considered to be a subgroup of the multiplicative group
of ℝ. In the completion K of the valued field k, the extension
of the valuation
is defined by
|x|=:limn→∞|xn|, |
when the Cauchy sequence x1,x2,…,xn,… of elements of k determines the element x of K.
Theorem.
The non-archimedean field k and its completion K have the same value group.
Proof. Of course, |k|⊆|K|. Let x=limn→∞xn 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
|xn-x|<|x| |
for all n>n0. For all these values of n we have
|xn|=|x+(xn-x)|=|x| |
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 |