On Rolle fields, with examples
type: Item
instance of: scholarly article
title: On Rolle fields, with examples
zbMATH Open document ID: 0820.12001
DOI: 10.1007/BF01197217
published in: Archiv der Mathematik
publication date: 1995-01-01T00:00:00Z
review text: As it is known, a field \(K\) is called a Rolle field if it admits a total order \(\leq\), compatible with the field structure and the following property is satisfied: if \(f\in K[X]\), \(a,b\in K\) and \(a<b\), \(f(a)= f(b)\), then there exists \(c\in K\) such that \(a<c<b\) and the derivative \(f'(c)= 0\). The paper focuses on henselian valued Rolle fields, concrete examples are provided by Rolle fields \(K= [[ k^{S,\leq} ]]\) of generalized power series, which have been studied in several papers of the author [with \textit{G. A. Elliott}, Arch. Math. 54, 365-371 (1990; Zbl 0688.13006) and Manuscr. Math. 75, 115-150 (1992; Zbl 0767.12001)]. Here \(S\) is a torsion-free abelian group, \(\leq\) is a compatible subtotal order, \(k\) is a field and \(K\) consists of all generalized power series with coefficients in \(k\), exponents in \(S\) and support which is artenian and narrow, i.e. quasi-well-ordered. A basic result is the analogue of the theorem of Brown, Craven and Pelling for henselian valued fields \((K,v)\): \(K\) is a Rolle field if and only if the residue field \(\overline {K}\) has the Rolle property and the value group is odd-divisible. This allows to give many concrete examples of Rolle fields of generalized power series (as indicated in the final section of the paper). In \S 3, one finds results concerning Rolle fields admitting only finitely many compatible total orders. As a consequence, it is shown how to obtain, for each \(n\geq 1\), examples of Rolle fields of generalized power series admitting exactly \(2^ n\) compatible total orders.
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 769519
zbMATH Keywords: henselian valued Rolle fields
zbMATH Keywords: generalized power series
author: Paulo Ribenboim
MaRDI profile type: Publication
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cites work: Q3356397
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cites work: Rings of generalized power series: Nilpotent elements
cites work: Fields: Algebraically closed and others
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description: scientific article; zbMATH DE number 769519