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Galois Cohomology and Class Field Theory

  • Textbook
  • © 2020


  • First textbook offering a complete exposition of local and global class field theory as well as arithmetic duality theorems
  • Provides the necessary background in Galois cohomology and homological algebra
  • Includes an appendix on analytical methods

Part of the book series: Universitext (UTX)

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Table of contents (18 chapters)

  1. Group Cohomology and Galois Cohomology: Generalities

  2. Local Fields

  3. Global Fields

  4. Duality Theorems


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.


“This book is a very good textbook for studying important roles of Galois cohomology in algebraic number theory … . This book consists of four parts, and the chapters are well constructed. … Examples and exercises are also well selected and interesting. This English translation is also very useful not only for students but also for researchers who study algebraic number theory in modern mathematical language, such as that of category theory or homological algebra.” (Yasushi Mizusawa, Mathematical Reviews, April, 2022)

“For students with no prior understanding of class field theory, the book is ideal. It is self-contained, and is based on a concatenation of master’s level courses given by the author. … his book seamlessly stitches together all the components in a neat and lucid manner. The whole process of learning this classical theory from Harari’s book makes it a painless and enjoyable experience. … The author takes a lot of care to make illuminating remarks in each chapter …” (Balasubramanian Sury, zbMATH 1466.11086, 2021)

Authors and Affiliations

  • Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, Orsay, France

    David Harari

About the author

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

  • Translated by: Andrei Yafaev

  • Series Title: Universitext

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2020

  • Softcover ISBN: 978-3-030-43900-2Published: 24 June 2020

  • eBook ISBN: 978-3-030-43901-9Published: 24 June 2020

  • 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

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