Primes of the Form x2 + ny2

Beschreibung

Modern number theory began with the work of Euler and Gauss tounderstand and extend the many unsolved questions left behind byFermat. In the course of their investigations, they uncovered newphenomena in need of explanation, which over time led to thediscovery of field theory and its intimate connection with complexmultiplication. While most texts concentrate on only the elementary or advancedaspects of this story, Primes of the Form x2 + ny2 begins withFermat and explains how his work ultimately gave birth to quadraticreciprocity and the genus theory of quadratic forms. Further, thebook shows how the results of Euler and Gauss can be fullyunderstood only in the context of class field theory. Finally, inorder to bring class field theory down to earth, the book exploressome of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can beexpressed in the form x2 + ny2. An incomplete answer is given usingquadratic forms. A better though abstract answer comes from classfield theory, and finally, a concrete answer is provided by complexmultiplication. Along the way, the reader is introduced to somewonderful number theory. Numerous exercises and examples areincluded. The book is written to be enjoyed by readers with modestmathematical backgrounds. Chapter 1 uses basic number theory andabstract algebra, while chapters 2 and 3 require Galois theory andcomplex analysis, respectively.

Autor*in

David A. Cox

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Mathematics Mathematik Number Theory Primzahl Zahlentheorie

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