The Beckman-Quarles theorem for rational spaces (Q1874368)

The Beckmann-Quarles theorem asserts that for \(d > 1\) every mapping from a \(d\)-dimensional real space \(R^d\) into itself preserving the unit distance (`unit-preserving', i.e. pairs of points with distance 1 will be mapped into pairs of points, again with distance 1) will preserve any distance and hence is an isometry. For the rational case \(Q^d\) the same proposition is not a foregone conclusion. For some numbers \(d\) the matter has already been settled; the author adds the proof for the case of dimensions \(d = 2 n^2-1\) for \(n > 2\). This proof is indeed versatile as it can also be applied to several other cases.