This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

This introduction to algebraic number theory discusses the classical concepts from the viewpoint of Arakelov theory. The treatment of class theory is particularly rich in illustrating complements, offering hints for further study, and providing concrete examples. It is the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available.

From the review: "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: Zentralblatt für Mathematik, 1992

This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

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.

Der Klassiker zum Thema bietet Lesern, die mit den Grundlagen der algebraischen Zahlentheorie vertraut sind, einen raschen Zugang zur Klassenkoerpertheorie. Die Neuauflage ist eine verbesserte Version des 1969 in der Reihe B. I.-Hochschulskripten (Bibliographisches Institut Mannheim) erschienenen gleichnamigen Bandes. Das Werk besteht aus drei Teilen: Im ersten wird die Kohomologie der endlichen Gruppen behandelt, im zweiten die lokale Klassenkoerpertheorie, der dritte Teil widmet sich der Klassenkoerpertheorie der endlichen algebraischen Zahlkoerper.

Algebraische Zahlentheorie: eine der traditionsreichsten und aktuellsten Grunddisziplinen der Mathematik. Das vorliegende Buch schildert ausf hrlich Grundlagen und H hepunkte. Konkret, modern und in vielen Teilen neu. Neu: Theorie der Ordnungen. Plus: die geometrische Neubegr ndung der Theorie der algebraischen Zahlk rper durch die "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt," die bis hin zum "Grothendieck-Riemann-Roch-Theorem" f hrt.