Algebraic Number Theory

• I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions and L-series.

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 fur Mathematik, 1992 "The book under review is the faithful and unabridged reprint of the original edition of J. Neukirch's excellent textbook on modern algebraic number theory ... . this unique classic in algebraic number theory is certainly of the highest advantage for new generations of students, teachers, and researchers in German-speaking mathematical communities, and therefore more than welcome. ... it will remain as one of the valuables in the legacy of an outstanding researcher and teacher in algebraic number theory forever." (Werner Kleinert, Zentralblatt MATH, Vol. 1131 (9), 2008)