Galois cohomology and class field theory. Translated from the French by Andrei Yafaev
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 in this language. The book, as I mentioned above, is self-contained but for two appendices. Some of the homological results needed in the main text are pushed to an appendix but then, this appendix contains also useful material on abelian categories and derived functors, etc., that a student could learn profitably for other purposes too. Also, a small amount of the analytic theory required in the proof of the first inequality of class field theory is covered in another appendix. The author takes a lot of care to make illuminating remarks in each chapter pointing to the fact that there are many approaches to the topic studied there and goes on to comment on advantages or disadvantages of the diverse approaches. It is also helpful that he states which results are used in the proof of a certain result. Each chapter has a set of very relevant but do-able exercises. To summarize, Harari's book is a modern exposition of a classical topic aimed at students beginning to learn class field theory. In my opinion, this book is the ideal go-to text for the uninitiated student to learn class field theory and duality theorems from, and appreciate the development of the subject in this process.
scientific article; zbMATH DE number 7188893