Integral-valued rational functions on valued fields
type: Item
instance of: scholarly article
title: Integral-valued rational functions on valued fields
zbMATH Open document ID: 0771.12003
DOI: 10.1007/BF02567653
author: Alexander Prestel
author: Cydara C. Ripoll
published in: Manuscripta Mathematica
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/155678
review text: The paper contains extracts from the second author's thesis written under the supervision of the first author. Let \(K\) be a field, \(v\) a non- trivial valuation ring of \(K\), \({\mathfrak m}\) the maximal ideal of \(v\), \((\widehat{K,v})\) the completion of the valued field \((K,v)\) as a uniform space, \(n\) a natural number. Denote by \(v[X]_{\text{sub}}:=\{f\in K[X]\mid f(v^ n)\subset v\}\) the ring of integral-valued polynomials and by \(v(X)_{\text{sub}}:=\{{g\over h}\mid g,h\in K[X]\) and \({g\over h}(a)\in v\) for all \(a\in v^ n\) such that \(h(a)\neq 0\}\) the ring of integral-valued rational functions. It is proved that for rational functions of one variable the identity \[ v(X)_{\text{sub}}=v[X]_{\text{sub}}(1+{\mathfrak m}v[X]_{\text{sub}})^{-1} \] holds iff the completion \((\widehat {K,v})\) of \((K,v)\) is locally compact or algebraically closed. Using a theorem from [\textit{P.-J. Cahen} and \textit{J. L. Chabert}, Bull. Sci. Math., II. Sér. 95, 295-304 (1971; Zbl 0221.13006)] the authors prove that the identity \(v(X)_{\text{sub}}=v[X](1+{\mathfrak m}v[X])^{-1}\) holds in any number of variables \(X=(X_ 1,\dots,X_ n)\) if the completion \((\widehat {K,v})\) is algebraically closed. Conversely, if this identity holds for \(n=1\), then \((\widehat {K,v})\) is algebraically closed.
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12J12
zbMATH DE Number: 60098
zbMATH Keywords: \(p\)-adic valuation
zbMATH Keywords: valuation ring
zbMATH Keywords: valued field
zbMATH Keywords: integral-valued polynomials
zbMATH Keywords: integral-valued rational functions
MaRDI profile type: Publication
cites work: Q3972040
cites work: Q5654135
cites work: Complete theories
cites work: Bemerkungen zur Theorie der formal \(p\)-adischen Körper
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description: scientific article; zbMATH DE number 60098