Galois module structure
type: Item
instance of: scholarly article
title: Galois module structure
zbMATH Open document ID: 0830.11042
published in: Fields Institute Monographs
publication date: 1995-01-25T00:00:00Z
review text: Over the last twenty years, the theory of Galois module structure has seen substantial progress. Originally it dealt with the study of \(G(L/ K)\)-modules \({\mathcal O}_L\) for a Galois extension \(L/K\) of global number fields. Fröhlich's conjecture states that the class of \({\mathcal O}_L\) in the class group \(Cl (\mathbb{Z}_K [G (L/K)])= K_0 (\mathbb{Z}[G(L/K)])/\{ \text{free modules}\}\) for a tamely ramified extension \(L/K\) coincides with the class of an analytic invariant \(W_{L/K}\) related with the Artin root number. It was \textit{M. J. Taylor} [Invent. Math. 63, 41-79 (1981; Zbl 0469.12003)]\ who proved this conjecture in 1981. Later T. Chinburg stated several generalizations of this conjecture using Chinburg invariants \(\Omega (L/ K,i)\). For example, Fröhlich-Chinburg's conjecture states that for a Galois extension \(L/K\) the second Chinburg invariant \(\Omega (L/K, 2)\in Cl( \mathbb{Z}[G(L/K) ])\) coincides with the class of \(W_{L/K}\). Very different from the book of \textit{A. Fröhlich} [Galois module structure of algebraic integers, Springer-Verlag (1983; Zbl 0501.12012)], this book is devoted to discussions of these conjectures and corresponding techniques. It is based on a graduate course given by the author at the Fields Institute in 1993. There are seven chapters and several dozen exercises. The first chapter contains many preliminaries in a brief form. In particular, a method of explicit Brauer induction, developed by the author and R. Boltje, is described there. It seems to be very useful in this subject in numerous applications throughout the book. The second chapter deals with the class group of an integral group-ring and the second Chinburg invariant. The Fröhlich-Chinburg conjecture is verified for some subextensions of cyclotomic extensions using \(p\)-adic \(L\)-functions. The third chapter introduces logarithmic technique which was originally discovered by M. J. Taylor and R. Oliver. Here the explicit Brauer induction provides a …
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 715354
zbMATH Keywords: Galois module structure
zbMATH Keywords: Fröhlich's conjecture
zbMATH Keywords: Artin root number
zbMATH Keywords: Chinburg invariants
zbMATH Keywords: Fröhlich-Chinburg's conjecture
zbMATH Keywords: several dozen exercises
zbMATH Keywords: explicit Brauer induction
zbMATH Keywords: class group of an integral group- ring
zbMATH Keywords: second Chinburg invariant
zbMATH Keywords: logarithmic technique
zbMATH Keywords: group-ring logarithm
zbMATH Keywords: method of canonical factorization
zbMATH Keywords: cohomological studies
author: Victor P. Snaith
reviewed by: I. B. Fesenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 715354