Topological characterization of p-adic numbers and an application to minimal Galois extensions.

*(English)*Zbl 0624.22004The 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.

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 |