Galois cohomology and class field theory /

Galois cohomology and class field theory /

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Bibliographic Details
Author / Creator:Harari, David, author.
Uniform title:Cohomologie galoisienne et théorie du corps de classes. English
Imprint:[Les Ulis, France] : EDP Sciences ; Cham : Springer. [2020]
©2020
Description:1 online resource (xiv, 338 pages) : illustrations.
Language:English
Series:Universitext, 2191-6675
Universitext.
Subject:
Galois cohomology.
Class field theory.
Galois cohomology.
Class field theory.
Number theory.
Electronic books.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12606155
Hidden Bibliographic Details
Other authors / contributors:Yafaev, Andrei, translator.
ISBN:3030439011
9783030439019
9783030439002
3030439003
Notes:Includes bibliographical references and index.
Description based on online resource; title from resource home page (ProQuest Ebook Central, viewed October 2, 2020).
Summary: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"--Publisher's description.
Other form:Print version: Harari, David. Galois cohomology and class field theory Cham : Springer, [2020] 9783030439002
Standard no.:10.1007/978-3-030-43
Table of Contents:
  • Part I. Group cohomology and Galois cohomology: generalities. Cohomology of finite groups: basic properties
  • Groups modified à la Tate, cohomology of cyclic groups
  • P-groups, the Tate-Nakayama theorem
  • Cohomology of profinite groups
  • Cohomological dimension
  • First notions of Galois cohomology
  • Part II. Local fields. Basic facts about local fields
  • Brauer group of a local field
  • Local class field theory: the reciprocity map
  • The Tate local duality theorem
  • Local class field theory: Lubin-Tate theory
  • Part III. Global fields
  • Basic facts about global fields
  • Cohomology of the idèles: the class field axiom
  • Reciprocity law and the Brauer-Hasse-Noether theorem
  • The abelianised absolute Galois group of a global field
  • Part IV. Duality theorems. Class formations
  • Poitou-Tate duality
  • Some applications
  • Appendices. Some results from homological algebra. Generalities on categories
  • Functors
  • Abelian categories
  • Categories of modules
  • Derived functors
  • Ext and tor
  • Spectral sequences
  • A survey of analytic methods
  • Dirichlet series
  • Dedekind [zeta] function; Dirichlet l-functions
  • Complements on the Dirichlet density
  • The first inequality
  • Class field theory in terms of ideals
  • Proof of the Čebotarev theorem.

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