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2020 | Book

Cohomology of Finite Groups: Basic Properties Read chapter Read first chapter

Galois Cohomology and Class Field Theory

Author: Prof. David Harari

Publisher: Springer International Publishing

Book Series : Universitext

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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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.

Table of Contents

Frontmatter

Group Cohomology and Galois Cohomology: Generalities

Frontmatter
Chapter 1. Cohomology of Finite Groups: Basic Properties
Abstract
This chapter gives the first properties of the cohomology of a finite group, which will be essential in the whole book.
David Harari
Chapter 2. Groups Modified à la Tate, Cohomology of Cyclic Groups
Abstract
This chapter introduces the modified Tate groups and cup-products.
David Harari
Chapter 3. p-Groups, the Tate–Nakayama Theorem
Abstract
This chapter proves the important Tate–Nakayama theorem, which will be used in local and global class field theory.
David Harari
Chapter 4. Cohomology of Profinite Groups
Abstract
This chapter extends all results of Chap. 1 to profinite groups.
David Harari
Chapter 5. Cohomological Dimension
Abstract
This chapter gives the main properties and examples of cohomological dimension.
David Harari
Chapter 6. First Notions of Galois Cohomology
Abstract
This chapter introduces Galois cohomology, the Brauer group, and their main properties.
David Harari

Local Fields

Frontmatter
Chapter 7. Basic Facts About Local Fields
Abstract
This chapter recalls basic facts on local fields.
David Harari
Chapter 8. Brauer Group of a Local Field
Abstract
This chapter starts local class field theory with the computation of the Brauer group of a local field and of its cohomological dimension.
David Harari
Chapter 9. Local Class Field Theory: The Reciprocity Map
Abstract
This chapter continues local class field theory with the reciprocity map and existence theorem via Kummer extensions.
David Harari
Chapter 10. The Tate Local Duality Theorem
Abstract
This chapter covers Tate local duality, unramified cohomology, and a second proof of the local existence theorem.
David Harari
Chapter 11. Local Class Field Theory: Lubin–Tate Theory
Abstract
This chapter covers local class field theory via Lubin–Tate construction, giving a third proof of the local existence theorem.
David Harari

Global Fields

Frontmatter
Chapter 12. Basic Facts About Global Fields
Abstract
This chapter recalls basic facts on global fields.
David Harari
Chapter 13. Cohomology of the Idèles: The Class Field Axiom
Abstract
This chapter starts global class field theory: cohomology of idèles, Kummer extensions, second inequality.
David Harari
Chapter 14. Reciprocity Law and the Brauer–Hasse–Noether Theorem
Abstract
This chapter continues global class field theory: reciprocity law, Brauer group, norm residue symbol.
David Harari
Chapter 15. The Abelianised Absolute Galois Group of a Global Field
Abstract
This chapter completes global class field theory: computation of the abelian Galois group, Hilbert class field, restricted ramification.
David Harari

Duality Theorems

Frontmatter
Chapter 16. Class Formations
Abstract
This chapter covers class formations and some homological algebra.
David Harari
Chapter 17. Poitou–Tate Duality
Abstract
This chapter covers the Poitou–Tate theorems and applications.
David Harari
Chapter 18. Some Applications
Abstract
This chapter gives a few complements on global fields (including computation of strict cohomological dimension).
David Harari
Backmatter
Metadata
Title
Galois Cohomology and Class Field Theory
Author
Prof. David Harari
Copyright Year
2020
Electronic ISBN
978-3-030-43901-9
Print ISBN
978-3-030-43900-2
DOI
https://doi.org/10.1007/978-3-030-43901-9

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