PlanetMath: complete ultrametric field

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complete ultrametric field

(Theorem)

A field equipped with a non-archimedean valuation $\vert\cdot\vert$ is called a non-archimedean field or also an ultrametric field, since the valuation induces the ultrametric $d(x,\,y) = \vert x\!-\!y\vert$ of .

Theorem 1 Let $(K,\,d)$ be a complete ultrametric field. A necessary and sufficient condition for the convergence of the series

$\displaystyle a_1\!+\!a_2\!+\!a_3\!+\ldots$

(1)

is that

$\displaystyle \lim_{n\to\infty}a_n = 0.$

(2)

Proof. Let $\varepsilon$ be any positive number. When (1) converges, it satisfies the Cauchy condition and therefore exists a number $m_\varepsilon$ such that surely

$\displaystyle \vert a_{m+1}\vert = \vert\sum_{j=1}^{m+1}a_j-\sum_{j=1}^{m}a_j\vert < \varepsilon$

for all $m \geqq m_\varepsilon$ ; thus (2) is necessary. On the contrary, suppose the validity of (2). Now one may determine such a great number $n_\varepsilon$ that

$\displaystyle \vert a_m\vert < \varepsilon \quad \forall m \geqq n_\varepsilon.$

No matter how great is the natural number

, the ultrametric then guarantees the inequality

$\displaystyle \vert a_m\!+\!a_{m+1}\!+\ldots+\!a_{m+n}\vert \leqq \max\{\vert a_m\vert,\,\vert a_{m+1}\vert,\,\ldots,\,\vert a_{m+n}\vert\} < \varepsilon$

always when $m \geqq n_\varepsilon$ . Thus the partial sums of (1) form a Cauchy sequence, which converges in the complete field. Hence the series (1) converges, and (2) is sufficient.

"complete ultrametric field" is owned by pahio.

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Also defines:

ultrametric field, non-archimedean field

Keywords:

convergence

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Cross-references: sufficient, complete, Cauchy sequence, partial sums, inequality, natural number, necessary, converges, number, positive, proof, series, necessary and sufficient, ultrametric, valuation, non-archimedean, field
There are 3 references to this entry.

This is version 11 of complete ultrametric field, born on 2005-01-03, modified 2008-04-23.
Object id is 6615, canonical name is CompleteUltrametricField.
Accessed 3218 times total.

Classification:

AMS MSC:	54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)
	12J10 (Field theory and polynomials :: Topological fields :: Valued fields)

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