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

Abstract

Let $S$ be a sphere in $\mathbf{R}^n$ whose center is not in $\mathbf{Q}^n$. We pose the following problem on $S$. \[ \text{``What is the closure of $S \cap \mathbf{Q}^n$ with respect to the Euclidean topology?''} \] In this paper we give a simple solution for this problem in the special case that the center $a = (a_i) \in \mathbf{R}^n$ of $S$ satisfies \[ \left\{ \sum_{i=1}^n r_i (a_i - b_i); \ r_1, \dots, r_n \in \mathbf{Q} \right\} = K \] for some $b = (b_i) \in S \cap \mathbf{Q}^n$ and some Galois extension $K$ of $\mathbf{Q}$. Our solution represents the closure of $S \cap \mathbf{Q}^n$ for such $S$ in terms of the Galois group of $K$ over $\mathbf{Q}$.