Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts) (英語) ペーパーバック – 2013/4/29
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An exciting approach to the history and mathematics of number theory
“. . . the author’s style is totally lucid and very easy to read . . .the result is indeed a wonderful story.” —Mathematical Reviews
Written in a unique and accessible style for readers of varied mathematical backgrounds, the Second Edition of Primes of the Form p = x2+ ny2 details the history behind how Pierre de Fermat’s work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. The book also illustrates how results of Euler and Gauss can be fully understood only in the context of class field theory, and in addition, explores a selection of the magnificent formulas of complex multiplication.
Primes of the Form p = x2 + ny2, Second Edition focuses on addressing the question of when a prime p is of the form x2 + ny2, which serves as the basis for further discussion of various mathematical topics. This updated edition has several new notable features, including:
• A well-motivated introduction to the classical formulation of class field theory
• Illustrations of explicit numerical examples to demonstrate the power of basic theorems in various situations
• An elementary treatment of quadratic forms and genus theory
• Simultaneous treatment of elementary and advanced aspects of number theory
• New coverage of the Shimura reciprocity law and a selection of recent work in an updated bibliography
Primes of the Form p = x2 + ny2, Second Edition is both a useful reference for number theory theorists and an excellent text for undergraduate and graduate-level courses in number and Galois theory.
DAVID A. COX, PhD, is William J. Walker Professor of Mathematics in the Department of Mathematics at Amherst College. Dr. Cox is the author of Galois Theory, Second Edition, also published by Wiley.
As a result of this style, the theory developed in this book is almost always shown for solving the problem's book.
"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.