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Topological characterization of p-adic numbers and an application to minimal Galois extensions. (English) Zbl 0624.22004
The author proves the following interesting results: (1) Let G be a nondiscrete locally compact group such that every non-trivial closed subgroup of G has finite index. Then G is isomorphic to the topological group of p-adic integers \({\mathbb{Z}}_ p\) for some prime p.
A Hausdorff topological group G is called minimal if it admits no coarser Hausdorff group topology. (2) Let G be an infinite minimal abelian group. Then every closed proper subgroup of G is open if and only if G can be embedded as a subgroup of \({\mathbb{Z}}_ p\) for some prime p.
(3) Let G be a locally compact nondiscrete noncompact abelian group. Then every proper closed subgroup of G is open if and only if G is isomorphic to the topological group of p-adic numbers \({\mathbb{Q}}_ p\) for some prime p iff any closed proper subgroup of G is compact.
Let K/k be an infinite Galois extension. It is called a minimal extension if every proper intermediate field is a finite extension of k. It is called a \(\Gamma\)-extension if G(K/k) is isomorphic to \({\mathbb{Z}}_ p\) for some prime p. (4) \(\Gamma\)-extensions are the only minimal Galois extensions.
Reviewer: T.Soundararajan

MSC:
22D05 General properties and structure of locally compact groups
22B05 General properties and structure of LCA groups
12F10 Separable extensions, Galois theory
11R18 Cyclotomic extensions
11R32 Galois theory
11S20 Galois theory
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