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Pure and Applied Mathematics: a Wiley Series of Texts, Monographs and Tracts Ser.: Primes of the Form X2 + Y2 : From Fermat to Class Field Theory and Complex Multiplication by David A. Cox (1989, Hardcover)
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About this product
Product Identifiers
PublisherWiley & Sons, Incorporated, John
ISBN-100471506540
ISBN-139780471506546
eBay Product ID (ePID)1505028
Product Key Features
Number of Pages351 Pages
LanguageEnglish
Publication NamePrimes of the Form X2 + Y2 : from Fermat to Class Field Theory and Complex Multiplication
Publication Year1989
SubjectNumber Theory
TypeTextbook
Subject AreaMathematics
AuthorDavid A. Cox
SeriesPure and Applied Mathematics: a Wiley Series of Texts, Monographs and Tracts Ser.
FormatHardcover
Dimensions
Item Height1.1 in
Item Weight24 Oz
Item Length9.6 in
Item Width6.3 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN89-005555
Dewey Edition19
Dewey Decimal512/.72
Table Of ContentFROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Field Theory and the Ccaron;ebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j-Invariants. The Class Equation. Ellpitic Curves. References. Index.
SynopsisNow available in paperback, this book explains how Fermat2s work gave birth to the quadratic reciprocity and the genus of quadratic forms. It goes on to show how the results of Euler and Gauss can be fully understood in the context of classical field theory and explores some of the magnificent formulas of complex multiplication., 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.
LC Classification NumberQA246.C69 1989
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