Der Satz von Benz-Rado. (The theorem of Benz-Rado) (Q1822050)

In the last few decades the characterization of isometries of quadratic spaces (also called metric vector spaces) has found considerable interest, see papers by A. D. Alexandrow, F. S. Beckman and D. A. Quarles, W. Benz, J. A. Lester and others. In the present paper the late author improves former results by W. Benz and F. Rado, proving the following theorem. Let K be a commutative field of characteristic not 2, 3, 5, and let \(K^ 2\) be the Minkowski plane over K, i.e. the quadratic space with the quadratic form xy in the cartesian coordinates x,y. Let furthermore \(f: K^ 2\to K^ 2\) be a mapping which preserves Minkowski distance 1, i.e. \[ (1)\quad \overline{XY}=1 \Rightarrow \overline{X^ fY^ f}=1\text{ for all points }X,Y. \] Then f is semiaffine. In particular, if (0,0) is fixed, then \[ (2)\quad (x,y)^ f= (x^{\alpha},y^{\alpha})\quad or\quad (y^{\alpha},x^{\alpha}), \] where \(\alpha\) is an isomorphism of K onto a subfield. The new and ingenious proof consists of the following steps. a) If distances 0 and 1 are preserved, then (2) holds. b) If distances 1 and 4 are preserved, then distance 0, too. c) If distance 1 is preserved, then distance 4, too. For the proof of c), the main part of the paper, the author constructs a sequence of 96 quadrangles of side lengths 1. He found this sequence with a computer, but it can easily be checked by hand. He also discusses exceptional cases with char \(K\in \{2,3,5\}\).