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.
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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 "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)
Language Notes
Text: English (translation)
Original Language: German
Original Language: German
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11 of 12 people found the following review helpful
By Jason Schorn
Format:Hardcover
After having no fun with Lang's text "Algebraic Number Theory" I began seking out something more complete and which was full of quality exposition. As a result of Amazon's approach to marketing towards members, I was recommended this book and decided quickly that I must have it. This book is marvelously well written, examples are kept to an un-overwhelming minimum, the problems are not trivial (at least to me) and in fact I feel this is the kind of book on par with, say, Paulo Ribenboim's "Classical Theory of Algebraic Numbers" since these are both the type of book you would want to take with you on a long trip or as Paulo says, "while stranded on a desert island". This book is by no means intended for those who are not fluent in both Number Theory as well as Algebra, both at the graduate level and obviously for those who are Mahematically gifted. I highly recommend this book to graduate students interested in Algebraic number theory as well as those needing a splendid reference.
9 of 10 people found the following review helpful
Format:Hardcover
This book is basically all you need to learn modern algebraic number theory. You need to know algebra at a graduate level (Serge Lang's Algebra) and I would recommend first reading an elementary classical Algebraic number theory book like Ian Stewart's Algebraic Number Theory, or Murty and Esmonde's Problem's in Algebraic Number theory.
Format:Hardcover|Amazon Verified Purchase
This book is great. It takes a different approach than some other texts I have recently read but is an excellent starting point for those interested in AN.
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