On the extensions of the ordering of a field to a simple transcendental field extension
type: Item
instance of: scholarly article
title: On the extensions of the ordering of a field to a simple transcendental field extension
zbMATH Open document ID: 0754.12003
DOI: 10.1007/BF01228282
author: Detlef Gröger
published in: Journal of Geometry
publication date: 1993-01-16T00:00:00Z
review text: It is well known that the orderings of the rational function field \(F(t)\) extending a given ordering \(<\) of the field \(F\) are in one-to-one correspondence to the Dedekind cuts in the real closure \(R\) of \((F,<)\). The author shows, that the group of fractional linear transformations of \(F\cup\{\infty\}\) acts on the set of cuts of \(R\), and that \((F(t),<_ 1)\) and \((F(t),<_ 2)\) are isomorphic, if and only if there exists a fractional linear transformation of \(F\cup\{\infty\}\) mapping the corresponding cuts of \(R\) onto each other. Some applications to the group of \(F\)-automorphisms of \((F(t),<)\) are given. The results presented here originate in the Ph.D. thesis of the author [Über angeordnete Fastkörper (Dissertation, Hannover) (1981); see also Result. Math. 5, 208-209 (1982; Zbl 0502.06007)] and complement work of \textit{H. Hofberger} [o-Automorphismen angeordneter Körper (Diplomarbeit, LMU München) (1986)] and \textit{D. Kijima} [Hiroshima Math. J. 17, 337-347 (1987; Zbl 0633.12012)]. For a more general approach see \textit{H. Hofberger} [Automorphismen formal reeller Körper (Dissertation, LMU München) (1991)].
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12F20
zbMATH DE Number: 92573
zbMATH Keywords: ordered fields
zbMATH Keywords: extension of orderings
zbMATH Keywords: transcendental extension
zbMATH Keywords: order preserving automorphism
zbMATH Keywords: Archimedean ordering
zbMATH Keywords: Dedekind cuts
zbMATH Keywords: real closure
zbMATH Keywords: fractional linear transformations
reviewed by: Franz B. Kalhoff
MaRDI profile type: Publication
cites work: Cuts of ordered fields
cites work: Q5590867
cites work: Q3223453
full work available at URL: https://doi.org/10.1007/bf01228282
OpenAlex ID: W182453231
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description: scientific article; zbMATH DE number 92573