Galois cohomology and class field theory. Translated from the French by Andrei Yafaev
type: Item
instance of: scholarly article
title: Galois cohomology and class field theory. Translated from the French by Andrei Yafaev
zbMATH Open document ID: 1466.11086
DOI: 10.1007/978-3-030-43901-9
author: David Harari
published in: Universitext
publication date: 2020-04-09T00:00:00Z
review text: It will be appropriate to talk about Harari's book along with another one on this subject by Georges Gras. Both of these books have been translated from the French originals recently and the translations are excellently done. \textit{G. Gras}'s book [Class field theory. From theory to practice. Berlin: Springer (2003; Zbl 1019.11032)] is very different from the present book by Harari in the following sense. Gras's book aims to introduce class field theory through examples and exercises. The main results of class field theory are not really proved in Gras's book and the stress is on understanding the results of class field through a plethora of explicit examples. For those who have already learnt the basics of class field theory, but are not yet adept at applying it, Gras's book is a wonderful source. The book by Harari is very different. It proves all the statements of local as well as global class field theory. The accent is on homological methods. In fact, a principal aim of the book is to present in complete detail the arithmetic Poitou-Tate duality theorems. 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. Hence, it is a tried and tested work. There are other books dealing with different aspects covered in this book -- such as the texts by Milne, Serre and Janusz on duality theorems, on Galois cohomology and class field theory, etc. Together these books perhaps cover the contents of Harari's book, but 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. Sticking to his sentiment that the cohomological method is the tour-de-force for class field theory and its duality theorems, Harari develops Group cohomology and Galois cohomology in complete detail and works out many of the proofs i…
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 7188893
zbMATH Keywords: Galois cohomology
zbMATH Keywords: local class field theory
zbMATH Keywords: global class field theory
zbMATH Keywords: Poitou-Tate duality
reviewed by: Balasubramanian Sury
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-43901-9
OpenAlex ID: W3037305167
Recommended article: Q4349107
Recommended article: Galois cohomology and class field theory
Recommended article: Q5562105
Recommended article: Q2756583
Recommended article: Q4133720
Recommended article: Q4039754
Recommended article: Field theory and the cohomology of some Galois groups
Recommended article: Q5615219
Recommended article: Galois Cohomology and Class Number in Constant Extension of Algebraic Function Fields
Recommended article: Galois theory and Lubin-Tate cochains on classifying spaces
description: scientific article; zbMATH DE number 7188893