An algebraic result on the topological closure of the set of rational points on a sphere whose center is non-rational. (English) Zbl 1069.14061

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


14P25 Topology of real algebraic varieties
12F10 Separable extensions, Galois theory
14G05 Rational points
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