On rational points of orthogonal group

TL;DR

This paper investigates the properties of rational vectors in orthogonal groups, demonstrating how certain rational vectors can be extended to rational orthonormal bases with specific integrality properties.

Contribution

It introduces a method to extend rational vectors with scaled integrality to full rational orthonormal bases sharing the same property.

Findings

Rational vectors with scaled integrality can be extended to orthonormal bases.

The constructed bases maintain the property of scaled integrality for all members.

The results contribute to understanding rational points in orthogonal groups.

Abstract

Let $n$ be a positive integer. We show that a unit rational space vector whose multiple by $n$ is an integer vector can be extended to a rational orthonormal basis whose all members have the same property.