On a problem of H. N. Gupta
type: Item
instance of: scholarly article
title: On a problem of H. N. Gupta
zbMATH Open document ID: 0873.51014
DOI: 10.1007/BF00150031
author: Andreas Blass
author: Victor V. Pambuccian
published in: Geometriae Dedicata
publication date: 1996-08-05T00:00:00Z
review text: The problem mentioned in the title is as follows. \textit{H. N. Gupta} [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 13, 551-552 (1965; Zbl 0139.14404)] provided an elementary axiomatization of finite-dimensional Cartesian spaces coordinatized by arbitrary ordered fields. In Can. Math. Bull. 12, 831-836 (1969; Zbl 0192.57302) he introduced axiom \(B\), stating that for any points \(x,y,z\) such that \(y\) is between \(x\) and \(z\) there is a right triangle having \(x\) and \(z\) as endpoints of the hypotenuse and \(y\) as the foot of the altitude of the hypotenuse, and axiom \(E\), which implies that the coordinate field is Euclidean. He asked the question: are axioms \(E\) and \(B\) equivalent when added to Euclidean geometry of arbitrary dimension \(n\geq 2\) over arbitrary ordered fields? It is easily seen that the answer is positive for \(n=2\), and \textit{W. Schwabhäuser} [Proc. Am. Math. Soc. 22, 233-234 (1969; Zbl 0177.23901)] showed that it is negative for \(n\geq 5\). The author of the present article shows that the answer is negative for \(n\geq 3\).
reviewed by: E. J. F. Primrose
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 913000
zbMATH Keywords: Euclidean geometry
zbMATH Keywords: arbitrary ordered fields
zbMATH Keywords: Cartesian spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00150031
OpenAlex ID: W2149432512
description: scientific article; zbMATH DE number 913000