Relative Galois structure of rings of integers.
type: Item
instance of: scholarly article
title: Relative Galois structure of rings of integers.
zbMATH Open document ID: 0641.12005
DOI: 10.1016/0022-314X(88)90065-0
author: Bouchaïb Sodaïgui
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Given a number field \(k\) and a finite abelian group \(\Gamma\); let \(\pi: \Gamma\simeq \text{Gal}(N/k)\) run over all realizations of \(\Gamma\) as Galois group over \(k\) with \(N/k\) at most tamely ramified. The problem of the realizable classes of \((k,\Gamma)\) asks to determine the subset of the class group of the group ring \(O_ k\Gamma\) of \(\Gamma\) over the ring of integers \(O_ k\) which are realized as \((O_ N)_{O_ k}\) where \(\pi\) runs as above; it is also of interest to impose additional `embedding problem' conditions on \(\pi\) [see for example p. 222--231 of the book by \textit{A. Fröhlich}, ``Galois module structure of algebraic integers.'' Berlin etc.: Springer (1983; Zbl 0501.12012) for an account of this idea]. We refer to the papers by \textit{L. R. McCulloh} (the most recent one appeared in [J. Reine Angew. Math. 375--376, 259--306 (1987; Zbl 0619.12008)]) for precise results on the realizable classes in the general situation. The paper under review is for a great part expository; some new results in special cases are given: For example, let \([k:{\mathbb Q}]=2\), \(\Gamma\simeq {\mathbb Z}/\ell\), \(\ell\) a prime number, and suppose that one considers the image of the realizable classes under the natural homomorphism \(Cl(O_ k\Gamma)\to Cl({\mathfrak M})\) where \({\mathfrak M}\) is the maximal order in \(k\Gamma\) ; the result is that this image does not become smaller if one imposes upon \(N\) the extra condition that the extension \(N/{\mathbb Q}\) is not normal.
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 13B05
zbMATH DE Number: 4043961
zbMATH Keywords: relative extensions of number fields
zbMATH Keywords: realizations as Galois group
zbMATH Keywords: realizable classes of class group
reviewed by: J. Brinkhuis
MaRDI profile type: Publication
cites work: Q3749140
cites work: Q5800391
cites work: Relative normal bases in some cyclotomic extensions
cites work: Arithmetic and Galois module structure for tame extensions.
cites work: Q5840258
cites work: Galois module structure of elementary abelian extensions
cites work: Q3312326
cites work: Q4402415
cites work: Critère de décomposition d'une extension de Kummer sur un sous-corps du corps de base
cites work: Q5628326
cites work: On Fröhlich's conjecture for rings of integers of tame extensions
cites work: S-groupes des classes d'un ordre arithmétique
cites work: Structures galoisienne des anneaux d'entiers d'extensions sauvagement ramifiees. I
cites work: Induced representations and projective modules
Recommended article: On arithmetically realizable classes
Recommended article: On Steinitz classes, realizable Galois module classes and embedding problems for non-abelian extensions of degree a power of 2
Recommended article: Realizable Galois module classes over the group ring for non abelian extensions
Recommended article: Steinitz classes of tamely ramified nonabelian extensions of odd prime power order
Recommended article: Structure galoisienne relative de la racine carrée de la codifférente d’extensions métacycliques non abéliennes
Recommended article: On realizable Galois module classes by the inverse different
Recommended article: Relative Galois module structure of octahedral extensions
Recommended article: Classes réalisables d'extensions non abéliennes
Recommended article: Realizable classes of nonabelian extensions of order \(p^3\)
Recommended article: Galois module structure of the square root of the inverse different over maximal orders
description: scientific article; zbMATH DE number 4043961