An algebraic result on the topological closure of the set of rational points on a sphere whose center is non-rational, II

TL;DR

This paper fully characterizes the topological closure of rational points on a sphere with a non-rational center, using algebraic automorphisms, extending previous partial results.

Contribution

It provides a complete algebraic description of the closure of rational points on spheres with non-rational centers, building on prior partial results.

Findings

Closure described via Q-automorphisms of algebraic closure

Complete characterization for non-rational centers

Extends previous partial results

Abstract

Let S be a sphere in R^n such that S\capQ^n\neq\emptyset and let Cl denote the closure operator in the Euclidean topology of R^n. If the center of S is in Q^n, then Cl(S\capQ^n) is S, as is easily proved. If the center of S is not in Q^n, then what is Cl(S\capQ^n)? This question, which was answered partially in the author's paper [Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 7, 146--149], is answered completely in this paper by representing Cl(S\capQ^n) in terms of the group of Q-automorphisms of the algebraic closure of Q(\gamma_1,...,\gamma_n) in C, where \gamma_1,...,\gamma_n denote the coordinates of the center of S.