Notes on Fermat's Last Theorem

The following is the publisher's blurb; read also 
what the critics say.

Alf van der Poorten, Notes on Fermat's Last Theorem
Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley--Interscience, January, 1996
222 + xvi pages
ISBN 0-471-06261-8
Library of Congress Call Number QA 244.V36 1996

An exciting introduction to number theory as reflected 
by the history of Fermat's Last Theorem

This book displays the unique talents of author Alfred J van der Poorten in mathematical exposition for mathematicians. Here, mathematics' most famous problem and the ideas underlying its recent solution are presented in a way that appeals to the imagination and leads the reader through related areas of mathematics. The first book to focus on Fermat's Last Theorem since Andrew Wiles presented his celebrated proof, Notes on Fermat's Last Theorem surveys 350 years of mathematical history in an amusing collection of tidbits, anecdotes, footnotes, exercises, limericks, references, illustrations, and more.

Proving that one can both read mathematics and read about it [Oops! I think this became "Providing that one can both read mathematics and read about it", in the blurbs on the book], this thoroughly accessible treatment

Helps students and professionals develop a background in number theory and provides introductions to the various fields of mathematics that are touched upon

Offers insight into the exciting world of mathematical research

Covers a number of areas appropriate for classroom use

Assumes only a year or so of university mathematics even for the more advanced topics

Explains why Fermat surely did not have the proof to his theorem

Examines the efforts of mathematicians over the centuries to solve the problem

Shows how the pursuit of the theorem contributed to the greater development of mathematics

Around 1637, the French jurist Pierre de Fermat scribbled in the margin of his copy of Diophantus' Arithmetica what came to be known as Fermat's Last Theorem, the most famous question in mathematical history. Stating that it is impossible to split a cube into two cubes, or a fourth power into two fourth powers, or any higher power into two like powers, but not leaving behind the marvelous proof he claimed to have had, Fermat prompted three and a half centuries of mathematical enquiry which culminated only recently with the proof of Fermat's Last Theorem by Andrew Wiles.

This book represents the first serious treatment of Fermat's Last Theorem since Wiles' proof. It is based on a series of lectures give by the author to celebrate Wiles' achievement, with each chapter explaining a separate area of number theory as it pertains to Fermat's Last Theorem. Together, they provide a concise history of the problem as well as a brief discussion of Wiles' proof and its implications. Requiring little more than one year of university mathematics and a liking for formulas, this overview provides many useful tips and cites numerous references for those who desire more mathematical detail.

The book's most distinctive feature is its easy-to-read, humorous style, complete with examples, anecdotes, and some of the less well known mathematics underlying the newly discovered proof. In the author's own words, the book deals with "serious mathematics without being too serious about it". Alf van der Poorten demystifies mathematical research, offers an intuitive approach to the subject --- loosely suggesting various definitions and unexplained facts --- and invites the reader to fill in the missing links in some of the mathematical claims.

Introduction

Biographical Remarks

Lecture I. Quasi-historical introduction

The cases $n=2$ and $n=4$. The Parisian Academy in the 1840 s. Notes: Some details. Descent. Algebraic numbers and integers. 1--10

Lecture II. Remarks on unique factorization

A digression. Notes: Continued fractions. Plagiarism 11--18

Lecture III. Elementary methods

Sophie Germain, Abel's formulas, Mirimanoff--Wieferich, ... Notes: Fermat's Theorem. Bernoulli numbers. Euler--Maclaurin. Pseudoprimes. Fermat numbers. Mersenne primes. Cranks. 19--30

Lecture IV. Kummer's arguments

Proof of the FLT for regular primes. Notes: Some remarks for undergraduates on elementary algebra. Equivalence relations. 31--39

Lecture V. Why do we believe Wiles? More quasi-history

Rantings. Work on the FLT this century. Notes: Euler's conjecture. The growing of the ``gap''. 41--50

Lecture VI. Diophantus and Fermat

What the study of diophantine equations is really all about. Notes: The chord and tangent method. Examples. 51--63

Lecture VII. A child's introduction to elliptic functions

For a precocious child. Notes: Discriminants. 65--73

Lecture VIII. Local and global

Some remarks on $p$-adic numbers. Notes: The Riemann $\zeta$-function. Much more on $p$-adic numbers. 75--88

Lecture IX. Curves

Particularly, about elliptic curves. Notes: Minimal model. Semisimplicity of the Frey curve. Birational equivalence. 89--101

Lecture X. Modular forms

Some formulas and assertions. Notes: More formulas. The discriminant function. 103--111

Lecture XI. The Modularity Conjecture

An attempt at an explanation. Notes: What's in a name? 113--122

Lecture XII. The functional equation

Poisson summation; $\vartheta$--functions. Notes: Details. Hecke operators. 123--133

Lecture XIII. Zeta functions and $L$--series

Introduction to the Birch--Swinnerton-Dyer Conjectures. Notes: Hasse's Theorem. 135--141

Lecture XIV. The ABC--Conjecture

Darmon and Granville's Generalized Fermat Equation. Notes: Hawkins primes. The Generalized Fermat Conjecture. 143--150

Lecture XV. Heights

Remarks on the Mordell--Weil Theorem. Notes: Lehmer's Question. Elliptic curves of high rank. 151--159

Lecture XVI. Class number of imaginary quadratic number fields

The proof of Goldfeld--Gross--Zagier. Notes: Composition of quadratic forms. Tate--Shafarevitch group. Jacobian. Heegner points. 161--176

Lecture XVII. Wiles' proof

Not the commutative algebra, of course. Notes: Some details. 177--188

Appendices

Appendix A. Remarks on Fermat's Last Theorem

For those who only want to pretend to have looked at the rest of this book. 187--199

Appendix B. "The Devil and Simon Flagg", by Arthur Porges

The devil fails where Wiles will succeed. 201--206

Appendix C. "Math Riots Prove Fun Incalculable", by Eric Zorn

Is the FLT truly as important as sport? 207--209

Index

211--222

Alf van der Poorten
1 Bimbil Place
Killara Australia 2071
Fax: +61 2 9850 8114
Mobile: +61 4 1826 3129
alf@math.mq.edu.au



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