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Primes of the Form x2+ny2 : Fermat, Class Field Theory, and Complex Multiplication.
An exciting approach to the history and mathematics of number theory ". . . the author's style is totally lucid and very easy to read . . .the result is indeed a wonderful story."--Mathematical ReviewsWritten in a unique and accessible style for readers of varied mathematical backgrou...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Hoboken :
Wiley,
2014.
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| Edición: | 2nd ed. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover ; Title Page ; Copyright ; Contents ; Preface to the First Edition ; Preface to the Second Edition ; Notation ; Introduction ; Chapter One: From Fermat to Gauss ; 1. Fermat, Euler and Quadratic Reciprocity ; A. Fermat ; B. Euler ; C.P = x2 + ny2 and Quadratic Reciprocity ; D. Beyond Quadratic Reciprocity ; E. Exercises ; 2. Lagrange, Legendre and Quadratic Forms ; A. Quadratic Forms ; B.P = x2 + ny2 and Quadratic Forms ; C. Elementary Genus Theory ; D. Lagrange and Legendre ; E. Exercises ; 3. Gauss, Composition and Genera ; A. Composition and the Class Group ; B. Genus Theory.
- C.P = x2 + ny2 and Euler''s Convenient Numbers D. Disquisitiones Arithmeticae ; E. Exercises ; 4. Cubic and Biquadratic Reciprocity ; A. Z[w] and Cubic Reciprocity ; B. Z[i] and Biquadratic Reciprocity ; C. Gauss and Higher Reciprocity ; D. Exercises ; Chapter Two: Class Field Theory ; 5. The Hilbert Class Field and P = x2 + ny2 ; A. Number Fields ; B. Quadratic Fields ; C. The Hilbert Class Field ; D. Solution of P = x2 + ny2 for Infinitely Many n ; E. Exercises ; 6. The Hilbert Class Field and Genus Theory ; A. Genus Theory for Field Discriminants.
- B. Applications to the Hilbert Class Field 7. Orders in Imaginary Quadratic Fields ; A. Orders in Quadratic Fields ; B. Orders and Quadratic Forms ; C. Ideals Prime to the Conductor ; D. The Class Number ; E. Exercises ; 8. Class Field Theory and the Cebotarev Density Theorem ; A. The Theorems of Class Field Theory ; B. The Čebotarev Density Theorem ; C. Norms and Ideles ; D. Exercises ; 9. Ring Class Fields and p = x2 + ny2 ; A. Solution of p = x2 + ny2 for All n ; B. The Ring Class Fields of Z[[square root]-27] and Z[[square root]-64] ; C. Primes Represented by Positive Definite Quadratic Forms.
- B. The Weber Functions C. J-invariants of Orders of Class Number 1 ; D. Weber''s Computation of J ([square root]-14) ; E. Imaginary Quadratic Fields of Class Number 1 ; F. Exercises ; 13. The Class Equation ; A. Computing the Class Equation ; B. Computing the Modular Equation ; C. Theorems of Deuring, Gross and Zagier ; D. Exercises ; Chapter Four: Additional Topics ; 14. Elliptic Curves ; A. Elliptic Curves and Weierstrass Equations ; B. Complex Multiplication and Elliptic Curves ; C. Elliptic Curves over Finite Fields ; D. Elliptic Curve Primality Tests ; E. Exercises ; 15. Shimura Reciprocity.