[PDF] Galois Cohomology And Class Field Theory eBook

Author : David Harari
Publisher : Springer Nature
Page : 336 pages
File Size : 50,64 MB
Release : 2020-06-24
Category : Mathematics
ISBN : 3030439011

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This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory. Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.

Author : Pierre Guillot
Publisher : Cambridge University Press
Page : 309 pages
File Size : 48,98 MB
Release : 2018-11
Category : Mathematics
ISBN : 1108421776

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A self-contained exposition of local class field theory for students in advanced algebra.

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 220 pages
File Size : 34,20 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1461210356

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Translation of the French Edition

Author : J. Neukirch
Publisher : Springer Science & Business Media
Page : 148 pages
File Size : 22,91 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 364282465X

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Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group theoretical principle, admitting an entirely elementary description. This de scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 215 pages
File Size : 22,66 MB
Release : 2013-12-01
Category : Mathematics
ISBN : 3642591418

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This is an updated English translation of Cohomologie Galoisienne, published more than thirty years ago as one of the very first versions of Lecture Notes in Mathematics. It includes a reproduction of an influential paper by R. Steinberg, together with some new material and an expanded bibliography.

Author : Georges Gras
Publisher : Springer Science & Business Media
Page : 517 pages
File Size : 16,92 MB
Release : 2013-11-11
Category : Mathematics
ISBN : 3662113236

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Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book.

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 249 pages
File Size : 26,22 MB
Release : 2013-06-29
Category : Mathematics
ISBN : 1475756739

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The goal of this book is to present local class field theory from the cohomo logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray.

Author : Helmut Koch
Publisher : Springer Science & Business Media
Page : 196 pages
File Size : 11,54 MB
Release : 2013-03-09
Category : Mathematics
ISBN : 3662049678

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Helmut Koch's classic is now available in English. Competently translated by Franz Lemmermeyer, it introduces the theory of pro-p groups and their cohomology. The book contains a postscript on the recent development of the field written by H. Koch and F. Lemmermeyer, along with many additional recent references.

Author : J. S. Milne
Publisher :
Page : 440 pages
File Size : 10,63 MB
Release : 1986
Category : Mathematics
ISBN :

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Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, ,tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Author : Philippe Gille
Publisher : Cambridge University Press
Page : 431 pages
File Size : 21,72 MB
Release : 2017-08-10
Category : Mathematics
ISBN : 1107156378

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The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results.