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Algebraic Number Fields: Second Edition
Gerald J. Janusz, University of Illinois, Urbana, IL
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Graduate Studies in Mathematics
1996; 276 pp; hardcover
Volume: 7
ISBN-10: 0-8218-0429-4
ISBN-13: 978-0-8218-0429-2
List Price: US$53
Member Price: US$42.40
Order Code: GSM/7
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The book is directed toward students with a minimal background who want to learn class field theory for number fields. The only prerequisite for reading it is some elementary Galois theory. The first three chapters lay out the necessary background in number fields, such as the arithmetic of fields, Dedekind domains, and valuations. The next two chapters discuss class field theory for number fields. The concluding chapter serves as an illustration of the concepts introduced in previous chapters. In particular, some interesting calculations with quadratic fields show the use of the norm residue symbol.

For the second edition the author added some new material, expanded many proofs, and corrected errors found in the first edition. The main objective, however, remains the same as it was for the first edition: to give an exposition of the introductory material and the main theorems about class fields of algebraic number fields that would require as little background preparation as possible. Janusz's book can be an excellent textbook for a year-long course in algebraic number theory; the first three chapters would be suitable for a one-semester course. It is also very suitable for independent study.

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Readership

Mathematics graduate students and faculty.

Reviews

"Gives a highly readable introduction into class field theory ... clearly written and may be recommended to everybody interested in the subject."

-- Zentralblatt MATH

"Provides a quick and self-contained introduction to the subject using only limited mathematical tools, hence it is accessible to a broader audience than most of the other texts on this topic."

-- Mathematical Reviews

Table of Contents

  • Subrings of fields (Chapter I)
  • Complete fields (Chapter II)
  • Decomposition groups and the Artin map (Chapter III)
  • Analytic methods and Ray classes (Chapter IV)
  • Class field theory (Chapter V)
  • Quadratic fields (Chapter VI)
  • Appendix
  • References
  • Index
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