Softcover ISBN:  9781470470289 
Product Code:  CHEL/387 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Electronic ISBN:  9781470471835 
Product Code:  CHEL/387.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 

Book DetailsAMS Chelsea PublishingVolume: 387; 2022; 533 ppMSC: Primary 11;
This book studies when a prime \(p\) can be written in the form \(x^{2} + ny^{2}\). It begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. After exploring cubic and biquadratic reciprocity, the pace quickens with the introduction of algebraic number fields and class field theory. This leads to the concept of ring class field and a complete but abstract solution of \(p = x^{2} + ny^{2}\). To make things more concrete, the book introduces complex multiplication and modular functions to give a constructive solution. The book ends with a discussion of elliptic curves and Shimura reciprocity. Along the way the reader will encounter some compelling history and marvelous formulas, together with a complete solution of the class number one problem for imaginary quadratic fields.
The book is accessible to readers with modest backgrounds in number theory. In the third edition, the numerous exercises have been thoroughly checked and revised, and as a special feature, complete solutions are included. This makes the book especially attractive to readers who want to get an active knowledge of this wonderful part of mathematics.ReadershipGraduate students and researchers interested in class field theory and complex multiplication.

Table of Contents

Chapters

From Fermat to Gauss

Class field theory

Complex multiplication

Additional topics

Solutions

References

Further reading

Index


Additional Material

Reviews

There are exercises throughout, including many cases of 'we leave the proof as an exercise.' The major change in this third edition is that full solutions are now included. Many of these were written by Roger Lipsett and then completed and revised by Cox. Inevitably, small errors and unclear spots were found in the course of preparing solutions, so one of the advantages of the new edition is that 'small errors have been fixed and many hints have been clarified and/or expanded.' The solutions to the exercises fill 219 pages of the book, almost doubling its size.
Fernando Guovea, Colby College, MAA Reviews


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This book studies when a prime \(p\) can be written in the form \(x^{2} + ny^{2}\). It begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. After exploring cubic and biquadratic reciprocity, the pace quickens with the introduction of algebraic number fields and class field theory. This leads to the concept of ring class field and a complete but abstract solution of \(p = x^{2} + ny^{2}\). To make things more concrete, the book introduces complex multiplication and modular functions to give a constructive solution. The book ends with a discussion of elliptic curves and Shimura reciprocity. Along the way the reader will encounter some compelling history and marvelous formulas, together with a complete solution of the class number one problem for imaginary quadratic fields.
The book is accessible to readers with modest backgrounds in number theory. In the third edition, the numerous exercises have been thoroughly checked and revised, and as a special feature, complete solutions are included. This makes the book especially attractive to readers who want to get an active knowledge of this wonderful part of mathematics.
Graduate students and researchers interested in class field theory and complex multiplication.

Chapters

From Fermat to Gauss

Class field theory

Complex multiplication

Additional topics

Solutions

References

Further reading

Index

There are exercises throughout, including many cases of 'we leave the proof as an exercise.' The major change in this third edition is that full solutions are now included. Many of these were written by Roger Lipsett and then completed and revised by Cox. Inevitably, small errors and unclear spots were found in the course of preparing solutions, so one of the advantages of the new edition is that 'small errors have been fixed and many hints have been clarified and/or expanded.' The solutions to the exercises fill 219 pages of the book, almost doubling its size.
Fernando Guovea, Colby College, MAA Reviews