On preordered fields related Hilbert's 17th problem
type: Item
instance of: scholarly article
title: On preordered fields related Hilbert's 17th problem
zbMATH Open document ID: 0691.12009
DOI: 10.1007/BF02571332
author: Guangxing Zeng
published in: Mathematische Zeitschrift
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174214
review text: This paper is a continuation of the author's earlier paper [Proc. Am. Math. Soc. 104, 335-342 (1988)]. In this paper, we continue to investigate preordered fields with the Weak Hilbert Property, for which the version of Hilbert's 17th problem on preordered fields is affirmative. By introducing another new notion ``strongly local density'', we show that a preordered field has the Weak Hilbert Property if and only if it is strongly locally dense. As a succedent result, we give again a characterization of preordered fields with the Weak Hilbert Property via special real places, which just generalizes the crucial implication in the definition of Weak Hilbert Property from rational to arbitrary function fields. Moreover, by the preceding result, we show the following result: Let both (K,S) and \((K^*,S^*)\) be preordered fields, and let \((K^*,S^*)\) be a finitely generated extension of (K,S). If (K,S) has the Weak Hilbert Property, then \((K^*,S^*)\) has also the Weak Hilbert Property. Finally, we point out the fact that the family of preordered fields with the Weak Hilbert Property is not inductive.
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 12J15
zbMATH DE Number: 4130498
zbMATH Keywords: weak Hilbert property
zbMATH Keywords: Hilbert's 17th problem
zbMATH Keywords: preordered fields
zbMATH Keywords: strongly local density
MaRDI profile type: Publication
cites work: Note on Artin’s solution of Hilbert’s 17th problem
cites work: Q3884218
cites work: The theory of real places
cites work: Q4129550
cites work: Q4141914
cites work: A Characterization of Preordered Fields with the Weak Hilbert Property
OpenAlex ID: W1979932231
description: scientific article; zbMATH DE number 4130498