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

An exciting approach to the history and mathematics of numbertheory

“. . . the author’s style is totally lucid and veryeasy to read . . .the result is indeed a wonderful story.”—Mathematical Reviews

Written in a unique and accessible style for readers of variedmathematical backgrounds, the Second Edition of Primes ofthe Form p = x2+ ny2details the history behind how Pierre de Fermat’s workultimately gave birth to quadratic reciprocity and the genus theoryof quadratic forms. The book also illustrates how results of Eulerand Gauss can be fully understood only in the context of classfield theory, and in addition, explores a selection of themagnificent formulas of complex multiplication.

Primes of the Form p = x2 +ny2, Second Edition focuses onaddressing the question of when a prime p is of the formx2 + ny2,which serves as the basis for further discussion of variousmathematical topics. This updated edition has several new notablefeatures, including:

• A well-motivated introduction to the classicalformulation of class field theory

• Illustrations of explicit numerical examples todemonstrate the power of basic theorems in various situations

• An elementary treatment of quadratic forms and genustheory

• Simultaneous treatment of elementary and advancedaspects of number theory

• New coverage of the Shimura reciprocity law and aselection of recent work in an updated bibliography

Primes of the Form p = x2 +ny2, Second Edition is both a usefulreference for number theory theorists and an excellent text forundergraduate and graduate-level courses in number and Galoistheory.