A discrete form of the Beckman—Quarles theorem for rational eight-space

Summary.

Let ${\Bbb Q}$ be the field of rationals numbers. We prove that: (1) if $x,y \in {\Bbb R}^{n}\,(n>1)$ and $|x - y|$ is constructible by means of ruler and compass then there exists a finite set $S_{xy}\subseteq {\Bbb R}^{n}$ containing x and y such that each map from S _xy to ${\Bbb R}^{n}$ preserving unit distance preserves the distance between x and y, (2) if $x,y \in {\Bbb Q}^{8}$ then there exists a finite set $S_{xy} \subseteq {\Bbb Q}^{8}$ containing x and y such that each map from S _xy to ${\Bbb R}^{8}$ preserving unit distance preserves the distance between x and y.