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On mappings of the Galois space

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Abstract

LetV be a metric vector space over a fieldK, dimV=n<∞, and let δ:V×V→K denote the corresponding distance function. Given a mappingσ:V→V such that δ(p,q) = 1⇒ δ(p σ,q ς) = 1, ifn=2, indV=1 and charK≠2, 3, 5, thenσ is semilinear [5], [11]; ifn≧3,K=R and the distance function is either Euclidean or Minkowskian, thenσ is linear [3], [10]. Here the following is proved: IfK=GF(p m),p>2 andn≧3, thenσ is semilinear (up to a translation), providedn≠0, −1, −2 (modp) or the discriminant ofV satisfies a certain condition. The proof is based on the condition for a regular simplex to exist in a Galois space, which may be of interest for its own sake.

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  1. Department of Mathematics, University of Cluj-Napoca, Str. KogĂlniceanu 1, Cluj-Napoca, Rumania

    F. Radó

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  1. F. Radó
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Radó, F. On mappings of the Galois space. Israel J. Math. 53, 217–230 (1986). https://doi.org/10.1007/BF02772860

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