**Graduate Studies in Mathematics**

Volume: 7;
1996;
276 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-0429-2

Product Code: GSM/7

List Price: $56.00

AMS Member Price: $44.80

MAA Member Price: $50.40

**Electronic ISBN: 978-1-4704-1141-1
Product Code: GSM/7.E**

List Price: $56.00

AMS Member Price: $44.80

MAA Member Price: $50.40

# Algebraic Number Fields: Second Edition

Share this page
*Gerald J. Janusz*

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.

#### Readership

Mathematics graduate students and faculty.

#### Reviews & Endorsements

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

# Table of Contents

## Algebraic Number Fields: Second Edition

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter I. Subrings of Fields 112 free
- 1. Localization 112
- 2. Integral Dependence 415
- 3. Discrete Valuation Rings and Dedekind Rings 718
- 4. Fractional Ideals and the Class Group 1627
- 5. Norms and Traces 1930
- 6. Extensions of Dedekind Rings 2536
- 7. Ramification and the Discriminant 3344
- 8. Norms of Ideals 4253
- 9. Algebraic Integers 4657
- 10. Cyclotomic Fields 5263
- 11. Quadratic Reciprocity 5970
- 12. Lattices in Real Vector Spaces 6273
- 13. The Class Number and the Unit Theorem 6778

- Chapter II. Complete Fields 8394
- Chapter III. Decomposition Groups and the Artin Map 121132
- Chapter IV. Analytic Methods and Ray Classes 135146
- Chapter V. Class Field Theory 169180
- 1. Cohomology of Cyclic Groups 169180
- 2. Preparations for the Second Inequality 172183
- 3. A Norm Index Computation 177188
- 4. The Fundamental Equality for Cyclic Extensions 184195
- 5. The Reciprocity Theorem 190201
- 6. Ideal Groups, Conductors, and Class Fields 199210
- 7. Reduction Steps toward the Existence Theorem 203214
- 8. Kummer Extensions and the S-unit Theorem 205216
- 9. The Existence Theorem 208219
- 10. Some Consequences of the Classification Theorem 215226
- 11. Norm Residues and the Conductor 218229
- 12. The Hilbert Class Field 228239

- Chapter VI. Quadratic Fields 233244
- Appendix 261272
- References 273284
- Index 275286
- Back Cover Back Cover1288