value group of completion

Let $k$ be a field and  $|\cdot |$  its non-archimedean valuation of rank one (http://planetmath.org/KrullValuation).  Then its value group  $|k\setminus \{0\}|$  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

$$|x|=:\underset{n\to \mathrm{\infty }}{lim}|x_n|,$$

when the Cauchy sequence  $x_1,x_2,\mathrm{\dots },x_n,\mathrm{\dots }$  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,  $|k|\subseteq |K|$.  Let  $x={lim}_{n\to \mathrm{\infty }}{x}_n$  be any non-zero element of $K$, where $x_j$’s form a Cauchy sequence in $k$.  Then there exists a positive number $n_0$ such that

for all  $n>n_0$.  For all these values of $n$ we have

$$|x_n|=|x+(x_n-x)|=|x|$$

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

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