UID:
almahu_9949697647602882
Umfang:
1 online resource (395 p.)
ISBN:
1-281-05040-7
,
9786611050405
,
0-08-047877-8
Originaltitel:
Invitation aux mathématiques de Fermat-Wiles.
Inhalt:
Assuming only modest knowledge of undergraduate level math, Invitation to the Mathematics of Fermat-Wiles presents diverse concepts required to comprehend Wiles' extraordinary proof. Furthermore, it places these concepts in their historical context.This book can be used in introduction to mathematics theories courses and in special topics courses on Fermat's last theorem. It contains themes suitable for development by students as an introduction to personal research as well as numerous exercises and problems. However, the book will also appeal to the inquiring and mathematically
Anmerkung:
Description based upon print version of record.
,
Front Cover; Invitation to the Mathematics of Fermat-Wiles; Copyright page; Contents; Foreword; Chapter 1. Paths; 1.1 Diophantus and his Arithmetica; 1.2 Translations of Diophantus; 1.3 Fermat; 1.4 Infinite descent; 1.5 Fermat's "theorem" in degree 4; 1.6 The theorem of two squares; 1.7 Euler-style proof of Fermat's last theorem for n = 3; 1.8 Kummer, 1847; 1.9 The current approach; Chapter 2. Elliptic functions; 2.1 Elliptic integrals; 2.2 The discovery of elliptic functions in 1718; 2.3 Euler's contribution (1753); 2.4 Elliptic functions: structure theorems
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2.5 Weierstrass-style elliptic functions2.6 Eisenstein series; 2.7 The Weierstrass cubic; 2.8 Abel's theorem; 2.9 Loxodromic functions; 2.10 The function p; 2.11 Computation of the discriminant; 2.12 Relation to elliptic functions Exercises and problems; Chapter 3. Numbers and groups; 3.1 Absolute values on Q; 3.2 Completion of a field equipped with an absolute value; 3.3 The field of p-adic numbers; 3.4 Algebraic closure of a field; 3.5 Generalities on the linear representations of groups; 3.6 Galois extensions; 3.7 Resolution of algebraic equations; Chapter 4. Elliptic curves
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4.1 Cubics and elliptic curves4.2 Bézout's theorem; 4.3 Nine-point theorem; 4.4 Group laws on an elliptic curve; 4.5 Reduction modulo p; 4.6 N-division points of an elliptic curve; 4.7 A most interesting Galois representation; 4.8 Ring of endomorphisms of an elliptic curve; 4.9 Elliptic curves over a finite field; 4.10 Torsion on an elliptic curve defined over Q; 4.11 Mordell-Weil theorem; 4.12 Back to the definition of elliptic curves; 4.13 Formulae; 4.14 Minimal Weierstrass equations (over Z); 4.15 Hasse-Weil L-functions; Chapter 5. Modular forms; 5.1 Brief historical overview
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5.2 The theta functions5.3 Modular forms for the modular group SL2(Z)/{I, -I}; 5.4 The space of modular forms of weight k for SL2(Z); 5.5 The fifth operation of arithmetic; 5.6 The Petersson Hermitian product; 5.7 Hecke forms; 5.8 Hecke's theory; 5.9 Wiles' theorem; Chapter 6. New paradigms, new enigmas; 6.1 A second definition of the ring Zp of p-adic integers; 6.2 The Tate module Tl (E); 6.3 A marvellous result; 6.4 Tate loxodromic functions; 6.5 Curves EA,B,C; 6.6 The Serre conjectures; 6.7 Mazur-Ribet's theorem; 6.8 Szpiro's conjecture and the abc conjecture
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Appendix: The origin of the elliptic approach to Fermat's last theoremBibliography; Index
,
English
Weitere Ausg.:
ISBN 0-12-339251-9
Sprache:
Englisch
