On mappings of the Galois space
type: Item
instance of: scholarly article
title: On mappings of the Galois space
zbMATH Open document ID: 0594.51013
DOI: 10.1007/BF02772860
author: F. Radó
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let V be an n-dimensional metric vector space over a field K, \(\delta\) : \(V\times V\to K\) a distance function. Let \(ind V\) be the Witt index, that is, the dimension of all maximal isotropic subspaces, of V. Given a map \(\sigma\) : \(V\to V\) such that \(\delta (p,q)=1\) implies \(\delta (p^{\sigma},q^{\sigma})=1\), if \(n=2\), \(ind V=1\) and \(char K\neq 2,3,5,\) then \(\sigma\) has been proven to be semilinear [\textit{W. Benz}, J. Geom. 17, 193-201 (1981; Zbl 0499.51005), and J. Geom. 18, 70-77 (1982; Zbl 0499.51006); the author, J. Geom. 21, 164-183 (1983; Zbl 0547.51005)]. If \(n\geq 3\), \(K={\mathbb{R}}\), and \(\delta\) is either euclidean or minkowskian (that is, the bilinear form related to \(\delta\) has signature n or n-2, respectively), then \(\sigma\) is known to be linear [\textit{W. Benz}, Arch. Math. 34, 550-559 (1980; Zbl 0446.51015); \textit{J. Lester}, C. R. Math. Acad. Sci., Soc. R. Can. 3, 59-61 (1981; Zbl 0474.51024)]. The author now proves: If \(K=GF(p^ m)\), \(p>2\) and \(n\geq 3\), then \(\sigma\) is semilinear up to a translation, provided \(n\not\equiv 0,-1,-2(mod p)\) or the discriminant of V satisfies a certain condition. The proof is based on the conditions for regular simplexes to exist in a Galois space.
Mathematics Subject Classification ID: 51F10
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 51N25
Mathematics Subject Classification ID: 83A05
Mathematics Subject Classification ID: 51K99
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 3955858
zbMATH Keywords: metric vector space
zbMATH Keywords: distance function
zbMATH Keywords: Witt index
zbMATH Keywords: isotropic subspaces
zbMATH Keywords: regular simplexes
zbMATH Keywords: Galois space
Wikidata QID: Q29306645
MaRDI profile type: Publication
cites work: Q3241211
cites work: On Isometries of Euclidean Spaces
cites work: Eine Beckman-Quarles-Charakterisierung der Lorentztransformationen des \(\mathbb{R}^n\)
cites work: A Beckman Quarles type theorem for plane Lorentz transformations
cites work: On mappings preserving a single Lorentz-Minkowski-distance. III
cites work: Q3736436
cites work: A Beckman-Quarles type theorem for finite Desarguesian planes
cites work: Q5806279
cites work: Q5814341
cites work: Q3930331
cites work: A characterization of the semi-isometries of a Minkowski plane over a field K
cites work: Mappings of Galois planes preserving the unit Euclidean distance
cites work: Der Satz von Benz-Rado. (The theorem of Benz-Rado)
cites work: Zur Kennzeichnung distanztreuer Abbildungen in nichteuklidischen Räumen
full work available at URL: https://doi.org/10.1007/bf02772860
OpenAlex ID: W2052210201
description: scientific article; zbMATH DE number 3955858