Anneaux d'entiers dans le même genre
type: Item
instance of: scholarly article
title: Anneaux d'entiers dans le même genre
zbMATH Open document ID: 0552.12005
author: Jacques Queyrut
published in: Illinois Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \({\bar {\mathbb{Q}}}\) denote an algebraic closure of \({\mathbb{Q}}\), the field of rational numbers. If N is a finite extension of \({\mathbb{Q}}\), then \(G_ N\) denotes the Galois group Gal(\({\bar {\mathbb{Q}}}/N)\) and \({\mathbb{Z}}_ N\) denotes the ring of integers of N. For a finite group \(\Gamma\), \(\epsilon\) (\(\Gamma)\) is the set of extensions N of \({\mathbb{Q}}\) such that \(\Gamma\) is isomorphic to a subgroup of the automorphism group of N over \({\mathbb{Q}}\). If \(N\in \epsilon (\Gamma)\) then \({\mathbb{Z}}_ N\) has a \({\mathbb{Z}}[\Gamma]\)-structure of rank \(r=[N^{\Gamma}:{\mathbb{Q}}].\) The following result (conjectured by Wilson) is proved: If N,N'\(\in \epsilon (\Gamma)\), such that \({\mathbb{Z}}_ N\) is locally isomorphic to \({\mathbb{Z}}_{N'}\), as \({\mathbb{Z}}[\Gamma]\)-modules, then there exists a \({\mathbb{Z}}[\Gamma]\)-module X of finite type and without \({\mathbb{Z}}\)-torsion and an isomorphism of \({\mathbb{Z}}[\Gamma]\)-modules between \({\mathbb{Z}}_ N\oplus X\cong {\mathbb{Z}}_{N'}\oplus X\).
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 3879039
zbMATH Keywords: Grothendieck group
zbMATH Keywords: Galois module structure
zbMATH Keywords: ring of algebraic integers
zbMATH Keywords: stable isomorphism
reviewed by: Jan Van Geel
MaRDI profile type: Publication
Recommended article: Q4087311
Recommended article: Q5602223
Recommended article: Q5779077
Recommended article: Q5522295
Recommended article: Q5720250
Recommended article: Q5603283
Recommended article: Q3845419
Recommended article: Anneaux henséliens
Recommended article: Q5536689
description: scientific article; zbMATH DE number 3879039