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Primes of the Form x + ny: Fermat, Class Field Theory, and Complex Multiplication [Paperback]

by David A. Cox
5.0 out of 5 stars  See all reviews (2 customer reviews)

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Book Description

May 8, 1997 0471190799 978-0471190790
Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of field theory and its intimate connection with complex multiplication.

While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication.

The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included.

The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively.


Editorial Reviews

From the Publisher

Provides a general solution to the question of which primes p can be expressed in the form x2 + ny2. Covered first are the special cases considered by Fermat, which involve only quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the mignificent formulas of complex multiplication.

From the Back Cover

Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of class field theory and its intimate connection with complex multiplication.

While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication.

The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included.

The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively.

Product Details

  • Paperback: 368 pages
  • Publisher: Wiley-Interscience (May 8, 1997)
  • Language: English
  • ISBN-10: 0471190799
  • ISBN-13: 978-0471190790
  • Product Dimensions: 9.3 x 6.2 x 0.8 inches
  • Shipping Weight: 1.3 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #1,947,164 in Books (See Top 100 in Books)

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6 of 8 people found the following review helpful
5.0 out of 5 stars Definitive text March 1, 2004
Format:Paperback
This is the definitive text on the theory of primes in the form x^2 + ny^2. Problems with simple formulation like this (which primes can be written in that form, and for what n?) make the class field theory interesting and worth understanding. Those with interest in prime numbers would greatly appreciate the topics presented. Recommended for those with a strong background in number theory and algebra.
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5.0 out of 5 stars but buy the second edition! January 3, 2014
By Frank
Format:Paperback
This is a great book. But buy the second edition, which is priced at a level which won't oblige you to remortgage your house.
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