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Invitation to the Mathematics of Fermat-Wiles [Kietas viršelis]

3.71/5 (14 ratings by Goodreads)
(Université de Caen, Basse-Normandie, France)
  • Formatas: Hardback, 400 pages, aukštis x plotis: 244x171 mm, weight: 910 g
  • Išleidimo metai: 24-Sep-2001
  • Leidėjas: Academic Press Inc
  • ISBN-10: 0123392519
  • ISBN-13: 9780123392510
Kitos knygos pagal šią temą:
Invitation to the Mathematics of Fermat-WilesDidesnis paveikslėlis
  • Formatas: Hardback, 400 pages, aukštis x plotis: 244x171 mm, weight: 910 g
  • Išleidimo metai: 24-Sep-2001
  • Leidėjas: Academic Press Inc
  • ISBN-10: 0123392519
  • ISBN-13: 9780123392510
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780123392510.html
Raktažodžiai:
For 350 years, mathematicians struggled with Fermat's last theorem, until Andrew Wiles presented a proof in 1994. This text explains mathematical developments that helped lay the foundations for Wiles' proof. It sets the math in its historical context, then rigorously presents concepts required to grasp Wiles' proof, assuming a background in undergraduate math. Chapters cover paths, elliptic functions, numbers and groups, elliptic curves, and modular forms. This book is translated by Leila Schneps from the second edition of the French version, Invitation aux MathTmatiques de Fermat-Wiles , Dunod, 2001. Hellegouarch is affiliated with the University of Caen. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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.

Key Features
* 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.

Recenzijos

"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

Daugiau informacijos

Key Features * 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.
Foreword viii
Paths
1(67)
Diophantus and his Arithmetica
2(1)
Translations of Diophantus
2(1)
Fermat
3(1)
Infinite descent
4(3)
Fermat's ``theorem'' in degree 4
7(2)
The theorem of two squares
9(7)
A modern proof
10(2)
``Fermat--style'' proof of the crucial theorem
12(1)
Representations as sums of two squares
13(3)
Euler-style proof of Fermat's last theorem for n = 3
16(2)
Kummer, 1847
18(15)
The ring of integers of Q(ζ)
18(5)
A lemma of Kummer on the units of Z (ζ]
23(2)
The ideals of Z[ ζ]
25(1)
Kummer's proof (1847)
26(5)
Regular primes
31(2)
The current approach
33(35)
Exercises and problems
35(33)
Elliptic functions
68(50)
Elliptic integrals
68(3)
The discovery of elliptic functions in 1718
71(4)
Euler's contribution (1753)
75(2)
Elliptic functions: structure theorems
77(3)
Weierstrass-style elliptic Functions
80(5)
Eisenstein series
85(2)
The Weierstrass cubic
87(2)
Abel's theorem
89(3)
Loxodromic Functions
92(3)
The function p
95(2)
Computation of the discriminant
97(2)
Relation to elliptic Functions
99(19)
Exercises and problems
101(17)
Numbers and groups
118(54)
Absolute values on Q
118(5)
Completion of a field equipped with an absolute value
123(4)
The field of p-adic numbers
127(4)
Algebraic closure of a field
131(3)
Generalities on the linear representations of groups
134(6)
Galois extensions
140(9)
The Galois correspondence
141(2)
Questions of dimension
143(3)
Stability
146(1)
Conclusions
146(3)
Resolution of algebraic equations
149(23)
Some general principles
149(3)
Resolution of the equation of degree three
152(3)
Exercises and problems
155(17)
Elliptic curves
172(83)
Cubics and elliptic curves
172(7)
Bezout's theorem
179(4)
Nine-point theorem
183(2)
Group laws on an elliptic curve
185(4)
Reduction modulo p
189(3)
N-division points of an elliptic curve
192(3)
2-Division points
192(1)
3-Division points
193(1)
n-Division points of an elliptic curve defined over Q
194(1)
A most interesting Galois representation
195(2)
Ring of endomorphisms of an elliptic curve
197(5)
Elliptic curves over a finite field
202(3)
Torsion on an elliptic curve defined over Q
205(6)
Mordell-Weil theorem
211(1)
Back to the definition of elliptic curves
211(4)
Formulae
215(3)
Minimal Weierstrass equations (over Z)
218(5)
Hasse-Well L-Functions
223(32)
Riemann zeta function
223(1)
Artin zeta function
224(2)
Hasse-Weil L-function
226(2)
Exercises and problems
228(27)
Modular Forms
255(70)
Brief historical overview
255(5)
The theta Functions
260(14)
Modular forms for the modular group SL2 (Z)/{I-I}
274(15)
Modular properties of the Eisenstein series
274(6)
The modular group
280(7)
Definition of modular forms and functions
287(2)
The space of modular forms of weight k for SL2 (Z)
289(5)
The fifth operation of arithmetic
294(3)
The Petersson Hermitian product
297(2)
Hecke forms
299(5)
Hecke operators for SL2 (Z)
300(4)
Hecke's theory
304(4)
The Mellin transform
306(1)
Functional equations for the functions L (f, s)
307(1)
Wiles' theorem
308(17)
Exercises and problems
313(12)
New paradigms, new enigmas
325(34)
A second definition of the ring Zp, of p-adic integers
326(2)
The Tate module Te (E)
328(2)
A marvellous result
330(1)
Tate loxodromic functions
331(1)
Curves EA,B,C
332(4)
Reduction of certain curves EA,B,C
333(2)
Property of the field Kp associated to Eap, bp, cp
335(1)
Summary of the properties of Eap, bp, cp
335(1)
The Serre conjectures
336(3)
Mazur-Ribet's theorem
339(4)
Mazur-Ribet's theorem
340(1)
Other applications
341(2)
Szpiro's conjecture and the abc conjecture
343(16)
Szpiro's conjecture
343(1)
abc conjecture
344(1)
Consequences
344(4)
Exercises and problems
348(11)
Appendix: The origin of the elliptic approach to Fermat's last theorem 359(12)
Bibliography 371(4)
Index 375


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."