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 $\mathbf{Q}$.
Publié le : 2004-09-14
Classification:
Sphere,
rational point,
topological closure,
Galois group,
14G05,
14P25,
12F10
@article{1116442333,
author = {Matsushita, Jun-ichi},
title = {An algebraic result on the topological closure of the set of rational points on a sphere whose center is non-rational},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {80},
number = {6},
year = {2004},
pages = { 146-149},
language = {en},
url = {http://dml.mathdoc.fr/item/1116442333}
}
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., Tome 80 (2004) no. 6, pp. 146-149. http://gdmltest.u-ga.fr/item/1116442333/