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

by David A. Cox

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.

Genres: Mathematics, Textbooks

386 pages, Kindle Edition

First published September 1, 1989

About the author

David Archibald Cox (born September 23, 1948 in Washington, D.C.) is an American mathematician, working in algebraic geometry.

Cox graduated from Rice University with a Bachelor's degree in 1970 and his Ph.D. in 1975 at Princeton University, under the supervision of Eric Friedlander (Tubular Neighborhoods in the Etale Topology). From 1974 to 1975, he was assistant professor at Haverford College and at Rutgers University from 1975 to 1979. In 1979, he became assistant professor and in 1988 professor at Amherst College.

He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner basis), Torelli sets and toric varieties, and history of mathematics. He is also known for several textbooks. He is a fellow of the American Mathematical Society.

From 1987 to 1988 he was a guest professor at Oklahoma State University. In 2012, he received the Lester Randolph Ford Award for Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First.

Reviews

[Cyprien Saito · Rating 4 out of 5]

Well-written introduction to the study on the reciprocity law.