Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Jun-ichi Matsushita

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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 $\mathbf{Q}$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 80, Number 7 (2004), 146-149.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116442333

Digital Object Identifier
doi:10.3792/pjaa.80.146

Mathematical Reviews number (MathSciNet)
MR2094537

Zentralblatt MATH identifier
1069.14061

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

Keywords
Sphere rational point topological closure Galois group

Citation

Matsushita, Jun-ichi. An algebraic result on the topological closure of the set of rational points on a sphere whose center is non-rational. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 7, 146--149. doi:10.3792/pjaa.80.146. https://projecteuclid.org/euclid.pja/1116442333


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References

  • 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).)