PlanetMath: value group of completion

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value group of completion

(Theorem)

Let be a field and $\vert\cdot\vert$ its non-archimedean valuation of rank one. Then its value group $\vert k\!\smallsetminus\!\{0\}\vert$ may be considered to be a subgroup of the multiplicative group of $\mathbb{R}$ . In the completion of the valued field , the extension of the valuation is defined by

$\displaystyle \vert x\vert := \lim_{n\to\infty}\vert x_n\vert,$

when the Cauchy sequence $x_1,\,x_2,\,\ldots,\,x_n,\,\ldots$ of elements of

determines the element

Theorem 1 The non-archimedean field

and its completion

have the same value group.

Proof. Of course, $\vert k\vert \subseteq \vert K\vert$ . Let $x = \lim_{n\to\infty}x_n$ be any non-zero element of , where 's form a Cauchy sequence in . Then there exists a positive number such that

$\displaystyle \vert x_n-x\vert < \vert x\vert$

for all

. For all these values of

we have

$\displaystyle \vert x_n\vert = \vert x+(x_n-x)\vert = \vert x\vert$

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

"value group of completion" is owned by pahio.

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Also defines:

value group of the completion

Keywords:

rank one valuation

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Cross-references: ultrametric triangle inequality, positive, non-archimedean field, Cauchy sequence, extension, completion, multiplicative group, subgroup, value group, valuation, non-archimedean, field
There is 1 reference to this entry.

This is version 6 of value group of completion, born on 2005-01-27, modified 2006-12-26.
Object id is 6670, canonical name is ValueGroupOfCompletion.
Accessed 1589 times total.

Classification:

AMS MSC:	12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
	13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
	13J10 (Commutative rings and algebras :: Topological rings and modules :: Complete rings, completion)
	13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)

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