Galois Cohomology and Class Field Theory

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.

Preface.- Part I Group cohomology and Galois cohomology: generalities.- 1 Cohomology of finite groups.- 2 Cohomology of cyclic groups.- 3 p-groups, the Tate-Nakayama theorem.- 4 Cohomology of profinite groups.- 5 Cohomological dimension.- 6 First notions of Galois cohomology.- Part II Local fields.- 7 Basic facts about local fields.- 8 Brauer group of a local field.- 9 Local class field theory: the reciprocity law.- 10 The Tate local duality theorem.- 11 Local class field theory: Lubin-Tate theory.- Part III Global fields.- 12 Basic facts about global fields.- 13 Cohomology of the idèles.- 14 Reciprocity law.- 15 The abelianized absolute Galois group of a global field.- Part IV Duality theorems.- 16 Class formations.- 17 Poitou-Tate duality.- 18 Some applications.- Appendix.- A Some results from homological algebra.- B A survey of analytic methods.- References.- Index.

"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)