Book 34

Primes of the Form x2 + ny2

by David A Cox

Published 24 October 2011
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.

Book 61

Galois Theory

by David A Cox

Published 21 September 2004
An introduction to one of the most celebrated theories of mathematics Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox's Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics. Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel's theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami.
Anyone fascinated by abstract algebra will find careful discussions of such topics as:* The contributions of Lagrange, Galois, and Kronecker* How to compute Galois groups* Galois's results about irreducible polynomials of prime or prime-squared degree* Abel's theorem about geometric constructions on the lemniscate With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike.

Book 119

Primes of the Form x2+ny2

by David A Cox

Published 28 September 1989
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.

This set features: Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication (978-0-471-19079-0) and Galois Theory (978-0-471-43419-1) both by David A. Cox.