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$|k\!\smallsetminus\!\{0\}|$  may be considered to be a subgroup of the multiplicative group of $\mathbb{R}$.  In the completion $K$ of the valued field $k$, the extension of the valuation is defined by

 $|x|\;=:\;\lim_{n\to\infty}|x_{n}|,$

when the Cauchy sequence$x_{1},\,x_{2},\,\ldots,\,x_{n},\,\ldots$  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  $x=\lim_{n\to\infty}x_{n}$  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

 $|x_{n}\!-\!x|\;<\;|x|$

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