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# Invitation to the Mathematics of Fermat-Wiles

## 1st Edition

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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 informed reader intrigued by the unraveling of this fascinating puzzle.

@introbul:Key Features
@bul:* Rigorously presents the concepts required to understand Wiles' proof, assuming only modest undergraduate level math
* Sets the math in its historical context
* Contains several themes that could be further developed by student research and numerous exercises and problems
* Written by Yves Hellegouarch, who himself made an important contribution to the proof of Fermat's last theorem
* Written by Yves Hellegouarch, who himself made an important contribution to the proof of Fermat's last theorem.

Researchers, professors, and students in mathematics.

Foreword
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
1.6.1 A Modern Proof
1.6.2 “Fermat-Style” Proof of the Crucial Theorem
1.6.3 Representations as Sums of Two Squares
1.7 Euler-Style Proof of Fermat’s Last Theorem for n=3
1.8 Kummer, 1847
1.8.1 The Ring of Integers of Q(ξ)
1.8.2 A Lemma of Kummer on the Units of Z[ξ]
1.8.3 The Ideals of Z[ξ]
1.8.4 Kummer’s Proof (1847)
1.8.5 Regular Primes
1.9 The Current Approach
Exercises and Problems
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
2.5 Weierstrass-Style Elliptic Functions
2.6 Eisenstein Series
2.7 The Weierstrass Cubic
2.8 Abel’s Theorem
2.9 Loxodromic Functions
2.10 The Function ρ
2.11 Computation of the Discriminant
2.12 Relation to Elliptic Functions
Exercises and Problems
3 Numbers and groups
3.1 Absolute Values on Q
3.2 Completion of a Fequipped 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
3.6.1 The Galois Correspondence
3.6.2 Questions of Dimension
3.6.3 Stability
3.6.4 Conclusions
3.7 Resolution of Algebraic Equations
3.7.1 Some General Principles
3.7.2 Resolution of the Equation of Degree Three
Exercises and Problems
4 Elliptic Curves
4.1 Cubics and Elliptic Curves
4.2 B´ezout’s Theorem
4.3 Nine-Point Theorem
4.4 Group Laws on an

- No. of pages:
- 400

- Language:
- English

- Copyright:
- © 2002

- Published:
- 24th September 2001

- Imprint:
- Academic Press

- Print ISBN:
- 9780123392510

- Electronic ISBN:
- 9780080478777

Yves Hellegouarch studied at the École Normale Supérieure in Paris. He has been teaching at the University of Caen since 1970. In 1972 he wrote a thesis, "Elliptic Curves and Fermat's Equation."

Université de Caen, Basse-Normandie, France

"This text provides a sweeping introduction to all those mathematical topics, concepts, methods, techniques, and classical results that are necessary to understand Andrew Wiles's theory culminating in the first complete proof of Fermat's last theorem. The text is accessible, without compromising the rigor of its mathematical exposition, to reasoned undergraduate students, at least so for the most part it can serve as the basis for various teaching courses. It sets the whole discussion in a fascinating, generally educating historical context, thereby travelling - metaphorically speaking - through the centuries of mathematical history. No doubt, it is a true blessing that the English translation of this unique book is now at hand for a much wider public."**
Werner Kleinert (Berlin) in Zentralblatt MATH 1036**