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Sur une version algébrique de la notion de sous-groupe minimal relatif de \({\mathbb{R}}^ n\). (On an algebraic version of the notion of relative minimal subgroup of \({\mathbb{R}}^ n)\). (French) Zbl 0729.12014

In a previous paper [Acta Arith. 56, 257-269 (1990; Zbl 0672.10024)] in connection with arithmetic questions related to the strong approximation theorem in algebraic number theory, the author introduced the topological notion of “minimal dense” subgroup of \({\mathbb{R}}^ n\) : this is a finitely generated subgroup G of \({\mathbb{R}}^ n\), which is dense in \({\mathbb{R}}^ n\), and such that no subgroup of G of smaller rank is dense in \({\mathbb{R}}^ n\). This property can be stated in terms of the \({\mathbb{Q}}\)- vector subspace E of \({\mathbb{R}}^ n\) generated by G.
In the present paper the author replaces \({\mathbb{Q}}\subset {\mathbb{R}}\) by two fields \(k\subset K\), he introduces the algebraic notion of relative minimal subspaces of \(K^ n\) and the dual notion of relative star-like k-subspace of \(K^ n\), and he studies these new notions. He shows for instance that the dimension of a k-minimal subspace of \(K^ n\) is \(\leq 2n\) ; he also gives a description of the k-minimal subspaces of \(K^ n\) of dimension \(n+1\), 2n-1 or 2n. An example is as follows : a k-subspace E of \(K^ n\) of dimension 2n is minimal if and only if there exists a subfield F of K which is quadratic over k, and there exists a basis of \(K^ n\) over K, such that E is the F-subspace of \(K^ n\) which is generated by this basis.

MSC:

12F99 Field extensions
22E15 General properties and structure of real Lie groups
11R45 Density theorems
54H11 Topological groups (topological aspects)
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References:

[1] N. BOURBAKI . - Topologie générale, chap. VII . - Hermann, Paris, 1974 . · Zbl 0337.54001
[2] S. LANG . - Algebra . - Addison Wesley, Don Mills, Ontario, 1971 . MR 43 #3276 | Zbl 0848.13001 · Zbl 0848.13001
[3] D. ROY . - Sous-groupes minimaux des groupes de Lie commutatifs réels et applications arithmétiques , Acta arith., t. 56, 1990 , (à paraître). MR 92f:11098 | Zbl 0672.10024 · Zbl 0672.10024
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