THE BECKMAN-QUARLES THEOREM FOR RATIONAL d-SPACES, d EVEN AND d≥6

A. Tyszka [8] proved that every unit-distance preserving map­ ping of Q⁸ to Q⁸ is an isometry. The purpose of this paper is to extend this property to all even d, d ≥ 6, by using the result of Zaks [10]. A mapping f: Q^d-+ Q^d is called unit-distance preserving if llx - Yll = 1 implies llf (x) - f(y) ll = 1. The Beckman-Quarles Theorem ([1], see also [2]) states that every unit-distance preserving mapping of R^d to R^d, d ≥ 2, is an isometry. Tyszka [8] (see also [6, 7]) proved that every unit-distance preserving mapping of the rational 8-space Q⁸ to Q⁸ is an isometry. Zaks [10] extended Tyszka's result for unit-distance preserving mappings f: Q^d-+ Q^d, for all even d of the form d = 4k(k + 1), as well as for all odd d of the form 2k² - 1 which are also complete squares. The purpose of this paper is to extend the results of Tyszka and Zaks to all even dimensions d, d≥6.