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OverviewThis book studies when a prime $p$ can be written in the form $x^{2} + ny^{2}$. It begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. After exploring cubic and biquadratic reciprocity, the pace quickens with the introduction of algebraic number fields and class field theory. This leads to the concept of ring class field and a complete but abstract solution of $p = x^{2} + ny^{2}$. To make things more concrete, the book introduces complex multiplication and modular functions to give a constructive solution. The book ends with a discussion of elliptic curves and Shimura reciprocity. Along the way the reader will encounter some compelling history and marvelous formulas, together with a complete solution of the class number one problem for imaginary quadratic fields. The book is accessible to readers with modest backgrounds in number theory. In the third edition, the numerous exercises have been thoroughly checked and revised, and as a special feature, complete solutions are included. This makes the book especially attractive to readers who want to get an active knowledge of this wonderful part of mathematics. Full Product DetailsAuthor: David A. CoxPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.962kg ISBN: 9781470470289ISBN 10: 1470470284 Pages: 542 Publication Date: 30 December 2022 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsFrom Fermat to Gauss Class field theory Complex multiplication Additional topics Solutions References Further reading IndexReviewsAuthor InformationDavid A. Cox, Amherst College, MA. Tab Content 6Author Website:Countries AvailableAll regions |