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Class field theory

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The present manuscript is an improved edition of a text that first appeared under the same title in Bonner Mathematische Schriften, no.26, and originated from a series of lectures given by the author in 1965/66 in Wolfgang Krull's seminar in Bonn. Its main goal is to provide the reader, acquainted with the basics of algebraic number theory, a quick and immediate access to class field theory. This script consists of three parts, the first of which discusses the cohomology of finite groups. The second part discusses local class field theory, and the third part concerns the class field theory of finite algebraic number fields.

Mathematics --- Number theory --- Algebra --- algebra --- wiskunde --- getallenleer

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"The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further study, and concrete examples.... The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook.... The book is, without any doubt, the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available." W. Kleinert in Z.blatt f. Math., 1992 "The author's enthusiasm for this topic is rarely as evident for the reader as in this book. - A good book, a beautiful book." F. Lorenz in Jber. DMV 1995 "The present work is written in a very careful and masterly fashion. It does not show the pains that it must have caused even an expert like Neukirch. It undoubtedly is liable to become a classic; the more so as recent developments have been taken into account which will not be outdated quickly. Not only must it be missing from the library of no number theorist, but it can simply be recommended to every mathematician who wants to get an idea of modern arithmetic." J. Schoissengeier in Montatshefte Mathematik 1994.

Algebraic number theory --- Algebraïsche getallentheorie --- Nombres algébriques [Théorie des ] --- Algebraic number theory. --- 511.6 --- Number theory --- Algebraic number fields --- 511.6 Algebraic number fields --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra

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Number theory --- Algebraic fields --- Algebraic numbers --- Algebraïsche velden --- Corps algébriques --- Fields [Algebraic ] --- Galois [Theorie de ] --- Galois [Theorie van ] --- Galois theory --- Homologie --- Homology theory --- Théorie de Galois --- Algebraic fields. --- Galois theory. --- Homology theory. --- Corps algébriques --- Théorie de Galois --- Nombres, Théorie des. --- Number theory. --- Cohomologie. --- Galois cohomology --- Cohomologie galoisienne. --- Nombres, Théories des --- Cohomologie --- Cohomologie galoisienne --- Nombres, Théorie des --- Nombres algébriques, Théorie des

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