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) - Hardcover

Cox, David A.

Hardcover

ISBN 10: 0471506540 ISBN 13: 9780471506546

Publisher: Wiley–Blackwell, 1989

Synopsis

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.

About this title

Synopsis

Topics covered in this volume include: quadratic reciprocity, quadratic forms, genus theory, cubic and biquadratic reciprocity, class field theory, orders in quadratic fields, elliptic functions, modular functions and complex multiplication. The central question of when a prime p is of the form x2 + ny2 is covered by the author at both elementary and advanced level, thus making the book accessible to a wide range of readers.

About the Author

DAVID A. COX is Professor of Mathematics at Amherst College.

Edition details

• Publisher: Wiley–Blackwell
• Publication date: 1989
• Language: English
• ISBN 10: 0471506540
• ISBN 13: 9780471506546
• Binding: Hardcover
• Number of pages: 352