On some classes of Hilbertian fields
type: Item
instance of: scholarly article
title: On some classes of Hilbertian fields
zbMATH Open document ID: 0547.12016
DOI: 10.1007/BF03322485
author: Şerban A. Basarab
published in: Results in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The author introduces a new and important class of fields K, called ''strongly pseudo real closed'' (sprc), defined by the following property: Every field extension \(F| K\) which is at the same time totally real and totally transcendental, can be embedded over K into an elementary extension \({}^*K\) of K such that \({}^*K| F\) is totally transcendental. Recall that \(F| K\) is ''totally transcendental'' (tt) if K is algebraically closed in F. Also, \(F| K\) is ''totally real'' (tr) if every ordering of F can be extended to an ordering of K. [Warning: This terminology is not compatible with that of ''totally real number field'' in the usual sense of algebraic number theory.] The author's concept of sprc field has been inspired, on the one hand by \textit{A. Prestel}'s concept of pseudo real closed (prc) fields [Lect. Notes Math. 872, 127-156 (1981; Zbl 0466.12018)], and on the other hand by the Gilmore-Robinson characterization of Hilbertian fields [cf. the reviewer Lect. Notes Math. 498, 231-275 (1975; Zbl 0316.12103)]. Recall on the one hand, that K is prc if every tt and tr extension \(F| K\) can be K-embedded into an elementary extension \({}^*K\) of K. On the other hand, K is Hilbertian if Hilbert's irreducibility theorem holds over K; by Gilmore-Robinson this is the case iff the field K(t) of rational functions in one variable can be embedded over K into an elementary extension \({}^*K\) of K such that \({}^*K\) is tt over K(t). The object of this paper is to put into evidence that sprc fields do have interesting properties, combining in some way the properties of Prestel's prc fields with those of Hilbertian fields. The class of sprc fields can be recursively axiomatized in the ordinary language L of fields, \(L=(0,1,+,-,\cdot).\) The non-real sprc fields are model complete with respect to a certain natural extension by definitions L' of L. This extension is constructed in such a way that it admits as morphisms between fields precisely the tt extensions. Spe…
reviewed by: Peter Roquette
Mathematics Subject Classification ID: 12F99
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 11S99
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 3871508
zbMATH Keywords: recursively axiomatized class
zbMATH Keywords: pseudo real closed fields
zbMATH Keywords: strongly pseudo real closed
zbMATH Keywords: totally transcendental
zbMATH Keywords: totally real
zbMATH Keywords: Hilbertian fields
zbMATH Keywords: Hilbert's irreducibility theorem
zbMATH Keywords: model complete
zbMATH Keywords: model companionable
zbMATH Keywords: elimination of quantifiers
zbMATH Keywords: decidable
zbMATH Keywords: orderings
zbMATH Keywords: Nullstellensätze
zbMATH Keywords: function field
zbMATH Keywords: holomorphy ring
zbMATH Keywords: Prüfer ring
zbMATH Keywords: generalized Jacobson ring
zbMATH Keywords: p-adically closed fields
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322485
OpenAlex ID: W2096366972
description: scientific article; zbMATH DE number 3871508