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

Matsushita, Jun-ichi

Proc. Japan Acad., Ser. A 80, No. 7, 146-149 (2004).

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 \(\mathbb{Q}\).

MSC:

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

References:

[1]	Bourbaki, N.: Elements of Mathematics. Algebra. Chapters 4-7. Springer, Berlin-Heidelberg-New York (1990). (Originally published as Algèbre. Chapitres 4 à 7. Lecture Notes in Mathematics, 864, Masson, Paris (1981).) · Zbl 0498.12001