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

Abstract

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