Galois Module Structure of the Integers of E - Extensions
Author : Martin John Taylor
Publisher :
Page : 162 pages
File Size : 25,77 MB
Release : 1977
Category : Galois theory
ISBN :
Author : Martin John Taylor
Publisher :
Page : 162 pages
File Size : 25,77 MB
Release : 1977
Category : Galois theory
ISBN :
Author : A. Fröhlich
Publisher : Springer Science & Business Media
Page : 271 pages
File Size : 32,91 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 3642688160
In this volume we present a survey of the theory of Galois module structure for rings of algebraic integers. This theory has experienced a rapid growth in the last ten to twelve years, acquiring mathematical depth and significance and leading to new insights also in other branches of algebraic number theory. The decisive take-off point was the discovery of its connection with Artin L-functions. We shall concentrate on the topic which has been at the centre of this development, namely the global module structure for tame Galois extensions of numberfields -in other words of extensions with trivial local module structure. The basic problem can be stated in down to earth terms: the nature of the obstruction to the existence of a free basis over the integral group ring ("normal integral basis"). Here a definitive pattern of a theory has emerged, central problems have been solved, and a stage has clearly been reached when a systematic account has become both possible and desirable. Of course, the solution of one set of problems has led to new questions and it will be our aim also to discuss some of these. We hope to help the reader early on to an understanding of the basic structure of our theory and of its central theme, and to motivate at each successive stage the introduction of new concepts and new tools.
Author : Gove Griffith Elder
Publisher :
Page : 224 pages
File Size : 25,84 MB
Release : 1993
Category :
ISBN :
Author : Albrecht Fröhlich
Publisher : Springer
Page : 282 pages
File Size : 22,54 MB
Release : 1983
Category : Mathematics
ISBN :
Author : Victor Percy Snaith
Publisher : American Mathematical Soc.
Page : 220 pages
File Size : 18,90 MB
Release : 1994-01-01
Category : Mathematics
ISBN : 9780821871782
This is the first published graduate course on the Chinburg conjectures, and this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions. The computation of Hom-descriptions is facilitated by Snaith's Explicit Brauer Induction technique in representation theory. In this way, illustrative special cases of the main results and new examples of the conjectures are proved and amplified by numerous exercises and research problems.
Author : Victor Percy Snaith
Publisher : American Mathematical Soc.
Page : 218 pages
File Size : 19,41 MB
Release : 1994
Category : Mathematics
ISBN : 082180264X
Galois module structure deals with the construction of algebraic invariants from a Galois extension of number fields with group $G$. This title addresses the Chinburg conjectures. It provides the background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions.
Author : Lindsay N. Childs
Publisher : American Mathematical Soc.
Page : 311 pages
File Size : 37,16 MB
Release : 2021-11-10
Category : Education
ISBN : 1470465167
Hopf algebras have been shown to play a natural role in studying questions of integral module structure in extensions of local or global fields. This book surveys the state of the art in Hopf-Galois theory and Hopf-Galois module theory and can be viewed as a sequel to the first author's book, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, which was published in 2000. The book is divided into two parts. Part I is more algebraic and focuses on Hopf-Galois structures on Galois field extensions, as well as the connection between this topic and the theory of skew braces. Part II is more number theoretical and studies the application of Hopf algebras to questions of integral module structure in extensions of local or global fields. Graduate students and researchers with a general background in graduate-level algebra, algebraic number theory, and some familiarity with Hopf algebras will appreciate the overview of the current state of this exciting area and the suggestions for numerous avenues for further research and investigation.
Author : Alfred Weiss
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 23,31 MB
Release : 1996
Category : Mathematics
ISBN : 0821802658
This text is the result of a short course on the Galois structure of S -units that was given at The Fields Institute in the autumn of 1993. Offering a new angle on an old problem, the main theme is that this structure should be determined by class field theory, in its cohomological form, and by the behaviour of Artin L -functions at s = 0. A proof of this - or even a precise formulation - is still far away, but the available evidence all points in this direction. The work brings together the current evidence that the Galois structure of S -units can be described. This is intended for graduate students and research mathematicians, specifically algebraic number theorists.
Author : Caiqun Xiao
Publisher :
Page : 38 pages
File Size : 46,9 MB
Release : 1997
Category :
ISBN :
Author : Ted Chinburg
Publisher : American Mathematical Soc.
Page : 167 pages
File Size : 13,32 MB
Release : 1989
Category : Algebraic number theory
ISBN : 0821824589
The main object of this memoir is to describe and, in some cases, to establish, new systems of congruences for the algebraic parts of the leading terms of the expansions of [italic]L-series at [italic lowercase]s = 0. If these congruences hold, together with a conjecture of Stark which states (roughly) that the ratio of the leading term to the regulator is an algebraic integer, then the main conjecture is true. The greater part of the memoir is devoted to the study of these systems of congruences for certain infinite families of quaternion extensions [italic]N/[italic]K (that is, [capital Greek]Gamma quaternion order 8). It is shown that such extensions can be constructed with specified ramification, and that various unit and class groups are calculable. This permits the verification of the congruences, and the main conjecture can be established for one such family of extensions.