On the Galois module structure over CM-fields
type: Item
instance of: scholarly article
title: On the Galois module structure over CM-fields
zbMATH Open document ID: 0757.11044
DOI: 10.1007/BF02567089
published in: Manuscripta Mathematica
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/155728
review text: For a given number field \(K\) and a given finite abelian group \(\Delta\), let us call \(R(K,\Delta)\) the set of all realizations of \(\Delta\) as a Galois group \(\text{Gal}(N/K)\) of a tame field extension \(N\) of \(K\). Question: to what extent are the elements of \(R(K,\Delta)\) characterized by their ``ramification'' and their ``Galois module structure''? The main result of this paper is that, for a given CM-field \(K\), the unramified realizations in \(R(K,\Delta)\) are invariant under the involution of \(R(K,\Delta)\) which is induced by complex conjugation on \(K\) and by inversion on \(\Delta\). Several applications are given. For example, if \(\Delta\) is of odd order and \(K\) is totally real, the answer to the above question is affirmative.
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 91467
zbMATH Keywords: tame realization
zbMATH Keywords: Galois module structure
zbMATH Keywords: CM-field
zbMATH Keywords: unramified realizations
author: J. Brinkhuis
MaRDI profile type: Publication
cites work: Q3749140
cites work: Normal integral bases and complex conjugation.
cites work: Galois module structure as the obstruction to a local-global principle
cites work: Unramified Abelian Extensions of CM -Fields and their Galois Module Structure
cites work: The Group of Unramified Kummer Extensions of Prime Degree
cites work: Q3966247
cites work: Bases normales, unités et conjecture faible de Leopoldt. (Normal bases, units, and the weak Leopoldt conjecture)
cites work: \(\mathbb Z_p\)-extensions of complex multiplication fields
cites work: On Vandiver's conjecture and \({\mathbb{Z}}_ p\)-extensions of \({\mathbb{Q}}(\zeta_{p^ n})\)
cites work: Galois module structure of abelian extensions.
cites work: On Fröhlich's conjecture for rings of integers of tame extensions
description: scientific article; zbMATH DE number 91467