On the structure of nonarchimedean exponential fields. I
type: Item
instance of: scholarly article
title: On the structure of nonarchimedean exponential fields. I
zbMATH Open document ID: 0820.12002
DOI: 10.1007/BF01375519
author: Salma Kuhlmann
published in: Archive for Mathematical Logic
publication date: 1995-09-11T00:00:00Z
review text: Given an ordered field \(K\), we are interested in the structure of its ordered additive group \((K,+, 0,<)\) and that of its ordered multiplicative group \((K^{>0}, \cdot, 1,>)\). To that end, we consider those groups as valued modules, for their natural (Krull) valuations. In section 2, we give a survey of the general theory on valued and ordered modules that we need. In section 3, we give lexicographic decompositions of \((K,+, 0,<)\), and of \((K^{>0}, \cdot, 1,<)\). We then proceed to compute the natural valuation and skeleton of \((K^{>0}, \cdot, 1,<)\) in terms of those of \((K,+, 0,<)\). We use this computation to provide necessary and sufficient conditions on the value group \(v(K)\) and residue field \(\overline {K}\), for the \(L_{\infty\omega}\)-equivalence of the above mentioned groups. In particular, we show that if \(K\) is root closed for positive elements, then its additive group of infinitesimals \(I_ v\) is always infinitary equivalent to its multiplicative group of 1-units \(1+I_ v\). We conclude that if \(K\) is countable, or if \(K\) is a power series field, then \(i_ v\) and \(I+I_ v\) are always isomorphic. We then apply the results to exponential fields. We describe \(v(K)\) in that case, and introduce the notion of an exponential group. Finally, if \(K\) is countable or a power series field, we derive necessary and sufficient conditions on \(v(K)\) and \(\overline {K}\) for \(K\) to be exponential. In the countable case, we get a structure theorem for \(v(K)\) and derive a construction method for all countable nonarchimedean exponential fields.
reviewed by: Salma Kuhlmann
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 16W99
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 769482
zbMATH Keywords: nonarchimedean ordered fields
zbMATH Keywords: infinitary equivalence
zbMATH Keywords: ordered field
zbMATH Keywords: ordered additive group
zbMATH Keywords: ordered multiplicative group
zbMATH Keywords: valued modules
zbMATH Keywords: valuations
zbMATH Keywords: ordered modules
zbMATH Keywords: exponential fields
zbMATH Keywords: countable nonarchimedean exponential fields
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 769482