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Book Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

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Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

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    Available in PDF - DJVU Format | Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication.pdf | Language: ENGLISH
    David A. Cox(Author)

    Book details


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.

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. 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.

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Book details

  • PDF | 368 pages
  • David A. Cox(Author)
  • Wiley-Interscience (May 8, 1997)
  • English
  • 6
  • Science & Math

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Review Text

  • By Peter Krumins on December 9, 2016

    This is the most fun math book that you've never heard of. It addresses the most interesting question ever:"Given a positive integer n, which primes p can be expressed in the form p=x²+ny² where x and y are integers?"Holy crap! That is so interesting. I was jumping when I got this book. What primes can be expressed in this form? Tell me! Tell me quick I want to know! Tell me! Tell me! Tell me! Tell me!This book answers this question completely and along the way you will encounter some remarkably rich areas of number theory. It starts with a historic overview of how this problem came to be and solution attempts to similar problems by Fermat, Euler, Lagrange, Legendre and Gauss. You'll learn about quadratic reciprocity and bridge the gap between between elementary number theory and class field theory. At the end of the book you'll learn how to find a constructive solution of p=x²+ny² using modular functions and elliptic curves.I get obsessed with problems like this and can spend weeks on end figuring out every tiny obscure detail. I've only read 1/5 of it so far and it's most excellent. It's not a textbook but has fun exercises and a bunch of insight connecting various branches of math that no other book connects so elegantly. As soon as I free up I'll power through the rest of it. I'll quit facebook, twitter, email and just read this book and ignore everyone. It will be just me and this book for a few weeks.I wish there were more math books like this. There are a couple more similar books but none are as exciting. I'll review those books later.I've placed this book #26 in my Top 100 Mathematics, Coding and Science books list. Google for >>catonmat top 100 math coding science books<< to find my list.

  • By Not on January 23, 2014

    The thing I love the most of this book is the constant pursuit of solving a particular problem. This make reading fun and exciting.As a result of this style, the theory developed in this book is almost always shown for solving the problem's book.

  • By Suhaimi Ramly on March 1, 2004

    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.

  • By Zhenchao Ge on April 10, 2015

    My adviser tell me this is the best book to begin class filed.

  • By Frank on January 3, 2014

    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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