## 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)

### Citations:

Zbl 0708.11033; Zbl 0672.10024
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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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