© 2020

Galois Cohomology and Class Field Theory


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Group Cohomology and Galois Cohomology: Generalities

    1. Front Matter
      Pages 1-1
    2. David Harari
      Pages 55-64
    3. David Harari
      Pages 65-78
    4. David Harari
      Pages 79-85
    5. David Harari
      Pages 87-95
  3. Local Fields

    1. Front Matter
      Pages 97-97
    2. David Harari
      Pages 99-107
    3. David Harari
      Pages 109-117
    4. David Harari
      Pages 131-143
  4. Global Fields

    1. Front Matter
      Pages 159-159
    2. David Harari
      Pages 161-175
  5. Duality Theorems

    1. Front Matter
      Pages 233-233

About this book


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.


Galois cohomology Class Field Theory Local fields Global fields Poitou-Tate duality Lubin-Tate formal group Brauer group Profinite groups

Authors and affiliations

  1. 1.Laboratoire de Mathématiques d'OrsayUniversité Paris-SaclayOrsayFrance

About the authors

David Harari is a professor at the Université Paris-Sud (Orsay). He is a specialist in arithmetic and algebraic geometry, author of 40 research papers in these fields.

Bibliographic information

  • Book Title Galois Cohomology and Class Field Theory
  • Authors David Harari
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-030-43900-2
  • eBook ISBN 978-3-030-43901-9
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XIV, 338
  • Number of Illustrations 48 b/w illustrations, 2 illustrations in colour
  • Additional Information English translation of the original French edition published by EDP Sciences, Les Ulis, 2017. Jointly published with EDP Sciences, Les Ulis, France
  • Topics Number Theory
  • Buy this book on publisher's site
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