Author: Victor Percy Snaith Publisher: American Mathematical Soc. ISBN: 9780821871782 Category : Mathematics Languages : en Pages : 220
Book Description
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: Alfred Weiss Publisher: American Mathematical Soc. ISBN: 0821802658 Category : Mathematics Languages : en Pages : 106
Book Description
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: A. Fröhlich Publisher: Springer Science & Business Media ISBN: 3642688160 Category : Mathematics Languages : en Pages : 271
Book Description
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: Lindsay N. Childs Publisher: American Mathematical Soc. ISBN: 1470465167 Category : Education Languages : en Pages : 311
Book Description
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: Andrew Schultz Publisher: ISBN: 9780549063520 Category : Languages : en Pages : 130
Book Description
The cohomology groups associated to the absolute Galois group of a field E encode a great deal of information about E, with the groups Hm(GE, mu p) being of classical interest. These groups are linked to the reduced Milnor K-groups KmE/pK mE = kmE by the Bloch-Kato conjecture. Using this conjecture when E/F is a Galois extension of fields with Gal(E/F) ≃ Z/pnZ for some odd prime p, and additionally assuming xi p ∈ E, we study the groups H m(GE, mup) as modules over the group ring Fp [Gal(E/F)]. When E/F embeds in an extension E'/F with Gal(E'/F) ≃ Z/pn+1 Z , we are able to give a highly stratified decomposition of Hm(GE, mup). This allows us to give a decomposition of the cohomology groups of a p-adic extension of fields. In general we are able to give a coarse decomposition of Hm(GE, mu p), showing that many indecomposable types do not appear in Hm(GE, mup). With an additional assumption about the norm map Nnn-1 : kmEn → k mEn-1, we strengthen this coarse decomposition to a highly stratified one.
Author: Victor P. Snaith Publisher: Birkhäuser ISBN: 3034882076 Category : Mathematics Languages : en Pages : 318
Book Description
This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993. The course was one of four associated with the 1993-94 Fields Institute programme, which I helped to organise, entitled "Artin L-functions". Published as [132]' the final chapter of the course introduced a manner in which to construct class-group valued invariants from Galois actions on the algebraic K-groups, in dimensions two and three, of number rings. These invariants were inspired by the analogous Chin burg invariants of [34], which correspond to dimensions zero and one. The classical Chinburg invariants measure the Galois structure of classical objects such as units in rings of algebraic integers. However, at the "Galois Module Structure" workshop in February 1994, discussions about my invariant (0,1 (L/ K, 3) in the notation of Chapter 5) after my lecture revealed that a number of other higher-dimensional co homological and motivic invariants of a similar nature were beginning to surface in the work of several authors. Encouraged by this trend and convinced that K-theory is the archetypical motivic cohomology theory, I gratefully took the opportunity of collaboration on computing and generalizing these K-theoretic invariants. These generalizations took several forms - local and global, for example - as I followed part of number theory and the prevalent trends in the "Galois Module Structure" arithmetic geometry.
Author: Victor Percy Snaith
Author: Alfred Weiss
Author: A. Fröhlich
Author: Lindsay N. Childs
Author: Frank A. Chemotti
Author: Bart De Smit
Author: Martin John Taylor
Author: Albrecht Fröhlich
Author: Andrew Schultz
Author: Victor P. Snaith