Primes of the Form x2 + ny2
Fermat, Class Field Theory, and Complex Multiplication
Seiten
2011
Wiley-Interscience (Hersteller)
978-1-118-03275-6 (ISBN)
Wiley-Interscience (Hersteller)
978-1-118-03275-6 (ISBN)
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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.
Provides a general solution to the question of which primes p can be expressed in the form x² + ny². Covered first are the special cases considered by Fermat, which involve only 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 mignificent formulas of complex multiplication.
Provides a general solution to the question of which primes p can be expressed in the form x² + ny². Covered first are the special cases considered by Fermat, which involve only 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 mignificent formulas of complex multiplication.
DAVID A. COX is Professor of Mathematics at Amherst College.
FROM 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 Fields Theory and the Cebotarev 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.
| Erscheint lt. Verlag | 31.10.2011 |
|---|---|
| Reihe/Serie | Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| ISBN-10 | 1-118-03275-6 / 1118032756 |
| ISBN-13 | 978-1-118-03275-6 / 9781118032756 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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