Product Description

If only the margin had been wider! For more than 300 years,mathematicians labored to crack the secret of Fermat's Last Theorem,without any success. Finally, in 1995, a Princeton-based mathematiciannamed Andrew Wiles solved the riddle. Amir Aczel's accountof this brainteaser and its solution is an irresistible read. And formathematical dolts--like myself, for instance--it includes a concise,profusely illustrated history of mathematical theory from the BronzeAge to our own fin-de-siecle. ... Read more

Customer Reviews (38)

A Marvellous Little Book
I was led to this book by one of Mr. Aczel's recent books "Descartes' Secret Notebook". While I have read a lot of books on physics and astronomy over the years (for enjoyment), I confess upfront that I had not - until Descartes' Secret Notebook - read any books about math (other than of course the usual panoply of math texts read much earlier in life). Like Descartes' Secret Notebook, this book - Fermat's Last Theorem - was enjoyable and entertaining to read. At just 137 pages in length, Mr. Aczel, in my opinion, did a very good job of mixing a basic explanation of the mathes involved (sufficient for an uninformed reader such as myself to appreciate the "mathematical context" of the story), while creatively weaving the basic story concerning the proof of the theorem with the various personalities that ultimately led to the proof. The book moves along swiftly and can be finished in a day or two. It is well worth the read. Well done Mr. Aczel.

PS. Descartes' Secret Notebook was equally entertaining and enjoyable.

Fermat creates his own legend
Fermat's Last Theorem undoubtly accelerated the development of modern mathematics. It can be argued that the pursuit of this egnimatic puzzle was the main factor behind the revolution oftheoretical mathematics, namely complex analysis, topology and group theory, in the 18th and 19th centuries. Due to his fame, his teasing challenge to future mathematicians created a intellectual furor that he might not even have anticipated. Many math societies in Europe, for hundreds of years traditionally offered large rewards for the solution, and naturally, this motivated thousands of starving mathematicians throughout the centuries to pursue the proof (Wiles eventually received $50K from the German Wolfskehl society). The list of the participants reads like a Who's Who of mathematics. Even when they failed, the work was often not in vain nor unheeded. And they all failed, until Wiles. Even the greatest, including Euler who greatly influenced the eventual solution, and many suspected, Gauss, who claimedthat " Fermat's Last Theorem has very little interest for me, for I could easily lay down a multitude of such propositions, which one could neither prove or disprove." In other words (in apparent bitterness),' don't waste my time.'
But one wonders if Fermat really had the proof. He certainly did not have knowledge (in the early 1600's) of modular elliptic curves in non-Euclidian space, which was conceived over the centuries, and became the basis of the solution. Fermat was known for his sly arrogance. Descarte described him as a "braggart" and Wallis called him "that damned Frenchman".[1] And the 'Last' theorem was not his only one. In fact he was known for these cunning claims, including that he could prove that 26 was the ONLY number that lay between two numbers that are roots, and that prime numbers of the form 4n+1 is ALWAYS the sum of TWO squares. But it was the 'Last' theorem that had the power to captivate because of its deceiving simplicity. A child could understand the problem, but the greatest minds were helpless in solving it!?!?
This was a fine book. The reader is introduced to number theory beginning from the time of the Phythagorean Brotherhood, and most the great characters since. Even the casual mathematician could appreciate the effort behindthis historical endeavor, culminating with Wile's stunning accomplisment , thanks to the authors elucidating style. I'm sure that Wiles would have paraphrased Newton, "If I have seen further than others, its because I stood on the shoulders of giants'.
[1] Singh, Fermat's Egnima p.40

A brief history of the search for the proof for Fermat's Last Theorem
Okay, I think I remember the media coverage that surrounded the completion of the proof of Fermat's Last Theorem back in the 90's.My mathematics background is limited to what was required for my engineering degree, and many of the more modern concepts covered here are far beyond my ability to wrap my mind around (this is further complicated by my complete lack of patience for details...I happier looking at things from the macro level, as it were).Parts of this slim volume were very interesting, especially the discussion of the role and development of math in antiquity.However, once the book moves into the more modern developments, especially those of the 20th century, the book lost some of its ability to hold my interest.I don't know if it was because I had trouble understanding the concepts, or if it is because the author seems to bounce from topic to topic, with only a cursory discussion of how they are related.Perhaps a more detailed discussion of the interplay between the various approaches and the people involved might have helped.Or maybe not.In any event, this book does provide an adequate overview of how the various mathematical tools that were necessary for the proof of Fermat's theorem were developed and brought together, and should satisfy those, like me, who are simply curious about why this proof took so long to formulate.

Interesting read!
Mr. Aczel is to be commended for a nice quick tour through mathematics and Fermat's Last Theorem; for a math book it is quite a page turner. Aczel's insight on the inner-world of world-class mathematicians is informative and entertaining---once again, for a math book a surprise. The publisher included diagrams and photos, and while a trade paperback, the quality was good enough to convey the story. If you're looking for a quick primer of math history, this book may be for you---as according to Aczel the roots of Fermat's Theorem has roots back to Pythagoras. Very good book and recommended.

For me useful
I think the most telling thing about this book is that
it is out of print and going used for $0.13 from sellers.
One thing you learn about mathematics books is:
if they are any good at all they hold their value, even used.
The major thing the author does is show contempt for the reader by
never giving any real equations. I find his end notes
probably the most useful and they also show that he had insufficient background to write this book. For me the use of the book is the reference
to the mathematicians who did the work historically.
I'm disgusted that he never wrote out the modular form
equation that is on what the whole proof rests.
For that he is a cheat as an author:
people aren't dumb and they don't need this level of dumb me down text. ... Read more

Product Description
This book is intended for amateurs, students and teachers.The author presents partial results which could be obtained with exclusively elementary methods.The proofs are given in detail, with minimal prerequisites.An original feature are the ten interludes, devoted to important topics of elementary number theory, thus making the reading of this book self-contained.Their interest goes beyond Fermat's theorem. The Epilogue is a serious attempt to render accessible the strategy of the recent proof of Fermat's last theorem, a great mathematical feat. ... Read more

Customer Reviews (5)

Title is misleading but a pretty good book
First, I give this book for 4 stars because the title is misleading. The background needed to understand this book is a major in mathematics at the undergrad level. If you were a math major and earned your degree several years ago, you may still have trouble following the book. It is for "amateurs" with a math degree and who are still "in touch" with the math they learnt. The book is however quite good; give a good historical narrative along with enough mathematics to satisfy the reader.

The problem is mathematics has gotten so much abstract and complicated during the last 50 years that it is impossible for someone not trained in the exact specialized field to follow what is going on. Since Fermat's last theorem has caught the public attention, people want to know what the fuss is all about. Alas, they really cannot understand it or even appreciate the mathematics in general without strong background in number theory, Galois theory, elliptic curves, and so on. All books that try to cater to the layman have to decide where to draw the line. If they water down the mathematics, then they really cannot explain how the proof was got satisfactorily; and if they throw in one too many equations, it becomes incomprehensible to many--even mathematicians not in that specialized field. My advice to the general public is to watch the video on Fermat's last theorem. I think for the layman visual media is better than a book. Just Google the excellent UKTV documentary on Fermat's last theorem.

Difficult book but great topic coverage
Solid coverage of proofs relating to Fermat's Last Theorem up to Kummer's Theory.You will find proofs for n=2, n=3, n=4, n=5, and n=7.Requires solid background in Algebraic Number Theory.For example, you should already have a good understanding of the Quadratic Law of Reciprocity, Quadratic Fields, and Congruences.If you don't, I recommend Elementary Number Theory for Congruences and the Quadratic Law of Reciprocity and Stark's An Introduction to Number Theory for Quadratic Fields.I would also recommend starting out with Edward's Book on Fermat's Last Theorem which includes detailed coverage of Kummer's Theory.

Great selection of material, difficult book
I find that this is a great book if you are an instructor or have a solid background in algebraic number theory.If you are unfamiliar with the Legendre Symbol, Gaussian integers, or the Law of Quadratic Reciprocity, you may wish to start out with a book such as Elementary Number Theory.If your are familiar with Algebraic Number Theory and wish to study in detail the Fermat Last Theorem proofs up to Kummer's Theory, this is a great book.I would recommend starting out with Edward's Book (Fermat's Last Theorem), for analysis of Euclid's proof of N=3.I found this very useful as an example of applications of Gaussian integers and Eisenstein integers.Ribenboim is one of the top experts about Fermat's Last Theorem and he is to praised for putting these beautiful proofs down.Even so, I would recommend purchasing other books to help explain this one.I found Stark's book very helpful in understanding Quadratic Fields.

"Amateur" mathematicians, that is !
If, like me, you were fascinated to hear that Fermat's so-called "last theorem" had been proven in 1995, then read Simon Singh or Amir Aczel's books popularizing the proof in outline, you probably wanted something more.

If, like me, you are a person who took some math in college, enjoys recreational mathematics books of the Douglas Hofstadter and Ian Stewart genre, and even sometimes picks up the odd number-theory book, you might consider yourself an "amateur."

If...if... this might seem like the book for you. I'd suggest that its not.

The mathematics in this book and its level of presentation was simply impenetrable by me. Not slow going... "no" going. That's frustrating to admit, but in a way fine, since it affirms of my admiration at a distance of the work that professional mathematicians do. I have seen many cited who state that Wiles' proof is simply beyond the ken of even 95% of working mathematicians. I believe this book must really be intended to serve some fraction of that group.Perhaps within the fold of mathematics these would consider themselves "amateurs". My two stars are offered only for them.

The book is simply not for the "lay" amateur. And Ribenboim's titling of it suggests that he does not even know that this lower caste, containing those of us who enjoy recreational mathematics and would describe ourselves as "amateurs", even exists. We know we exist as something mathematically distinct from the general population by the simple fact of the universally raised eyebrows that confront any mention of our interest in mathematics. Nevertheless, like any other species in a niche, we will have to continue to feed on a sparse supply of intellectual sustenance and learn to avoid the over-rich and indigestible fare of the higher forms.

Finally, if you haven't read Singh or Aczel I'd offer the former 5 stars and the latter 3 but recommend both. A truly fascinating story.

An excellant work, good for any serious study of FLT.
I am a math instructor and graduate student at PVAMU, and am working on a thesis detailing the history of attempts to prove and the Wiles proof of FLT. The text was easily readable and the proofs were very well done. I wasable to follow the logic and math of all the presented proofs very well. ... Read more

Product Description
xⁿ + yⁿ = zⁿ, where n represents 3, 4, 5, ...no solution

"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."

With these words, the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations.What came to be known as Fermat's Last Theorem looked simple; proving it, however, became the Holy Grail of mathematics, baffling its finest minds for more than 350 years.In Fermat's Enigma--based on the author's award-winning documentary film, which aired on PBS's "Nova"--Simon Singh tells the astonishingly entertaining story of the pursuit of that grail, and the lives that were devoted to, sacrificed for, and saved by it.Here is a mesmerizing tale of heartbreak and mastery that will forever change your feelings about mathematics.Amazon.com Review
When Andrew Wiles of Princeton University announced a solutionof Fermat's last theorem in 1993, it electrified the world ofmathematics. After a flaw was discovered in the proof, Wiles had towork for another year--he had already labored in solitude for sevenyears--to establish that he had solved the 350-year-old problem. SimonSingh's book is a lively, comprehensible explanation of Wiles's workand of the star-, trauma-, and wacko-studded history of Fermat's lasttheorem. Fermat's Enigma contains some problems that offer ataste of the math, but it also includes limericks to give a feelingfor the goofy side of mathematicians. ... Read more

Customer Reviews (266)

Wonderful book!
Simon Singh's talent is that he can write intriguing stories centered on advanced scientific or mathematical concepts that even the layman can appreciate. His decision to write a book on Fermat's theorem is spot on- although the proof of the theorem cannot be understood by more than 99.9% of humans on the planet, anyone who has gone to high school can probably understand the simple problem statement. Singh starts off in his usual style of introducing the problem along with its historical background. The early chapters constitute a very nice condensed history of mathematics and number theory in particular. The middle chapters explain the initial attempts to prove the theorem. The last few chapters are centered on Andrew Wiles and his single handed quest to find the proof. By this time, the mathematics involved are out of reach for most of us but even then Singh is able to keep the reader immersed and spellbound. The story of Wiles' success in proving the theorem with almost zero external assistance is amazing and uplifting. He has become one of my heroes. A great book and highly recommended.

Extraordinary book!
Fermat's Enigma - The story of a riddle that confounded the world's greatest minds for 358 years is science history book written by Simon Singh. Being a PhD graduate from Cambridge, Simon Singh has written this book with so much simplicity that even a person without any mathematical background can appreciate the content. This book is one of its kind as it addresses only one theorem from the start to end. It narrates the struggles Andrew Wiles had to undergo before proving the theorem completely in 1995. This book also gives an account of other mathematicians who were involved.

Fermat's last theorem is nothing but an extension of the Pythogoras theorem. Fermat claimed that there are no whole number (non-zero) solution (x,y,z) for the equation

for n>2 upto n=infinity. Fermat was a mischievous mathematician who rarely gave complete proof for his theorems. He, therefore, claimed that he has a complete complete for this theorem and died before disclosing the solution. The race for proving this theorem started soon after.

This book starts from around 500BC when Pythogorean brotherhood headed by Pythogoras were the front runners in the mathematics. They formed a close community and did not let any of their inventions to the society. Singh has nicely illustrated the incident where Pythogoras executed his own student for proving that there were irrational numbers. Pythogorean brotherhood get eventually destroyed by another student who had been ignored citing his incompetence.

This book has few more very interesting events like how Taniyama committed suicide before proving his own conjecture which would have eventually proved Fermat's last theorem. Also, there is an interesting account of Wolfskehl whose life was saved by Fermat's last theorem and the award instituted by him for the person who would solve Fermat's last theorem. Further, the book lucidly explains the complexity of the theorem itself and the various failure attempts in proving it.

The most stunning part of the book is related to Andrew Wiles, the Princeton Professor, who secludes himself from the research community and conferences to focus on achieving his childhood dream, i.e. to prove Fermat's last theorem. The logical reasoning as explained in the book which Wiles took in order to prove the theorem would appeal even to people who have limited exposure to rigorous mathematics.

The turmoils and pressures he faced after 1993 when it was found by reviewers that there was a small gap in his initial proof are really well narrated. All the key persons involved in the proof apart from Wiles like Nick Katz, Shimura, Ken Ribet, Richard Taylor and others explain the steps taken in their own words. This makes the book even more accessible and credible. The book concludes by giving an account of how Wiles finally releases his paper with the complete proof of Fermat's last theorem in 1995.

On the whole, I loved reading every page of this book. I just completed reading this amazing book on a classical theorem. Simon Singh is an amazing writer and keeps the reader engrossed throughout. This book is highly recommended and is a must read for anyone interested in learning about a great mathematical theorem and its proof. Even people with just high school mathematics knowledge would love this book. I am already into reading Simon Singh's other books: The Code Book and Big Bang.

A Fun Read Through History
In Fermat's Enigma, Simon Singh offers a compelling trek through the last 2500 years of mathematical development, always focusing on developments that would impact a mathematical conundrum - a proof to Fermat's Last Theorem. This is a delightful book, and I think even the mathematically challenged will find the tale fun and exciting. While I think Simon Singh's The Code Book far outstrips this one in material, quality and challenge for the reader, this book is well worth the read.

The book is aesthetically pleasing, having a wonderful font, important when text is intermingled with numbers and variables. Illustrations litter the pages, providing diagrams and representations of key concepts as well as the faces of major players in the story. Where math is required it clearly stands apart and is handled in a way that complements without weighing down the story. Beyond that, it is a quick read.

While a good portion of the story covers a wide span in history, much of the story is a limited biography of Andrew Wiles, the mathematician responsible for the final ingenuity that proved Fermat's Last Theorem. Simon also highlights the secrecy of the work of Wiles, similar to that of Fermat's own work - with Fermat coming out as the more devious of the two, in my opinion. But the story clearly shows that the solution to Fermat's challenge rests not on the work of one man, but many notable men and women throughout history who devoted years to this problem. However, as with any retelling of historical events, Singh offers a subjective view of cause and effect that is somewhat arguable, and sometimes even derails the story a bit.

I particularly enjoyed reading about the development of mathematics among the Greeks, and the mathematical developments of the 1800's. Though the former does not speak to Fermat's Last Theorem, it provides the backdrop to it in the Pythagorian Theorem. And the latter provides an interesting look into the advances in both math and science that would shape the modern world. While Fermat's Last Theorem itself is not an "important" mathematical principal, its proof has been the catalyst for the integration of disparate areas of mathematics, as well as an impetus for developing new means of calculation and logic.

Not yet received
I ordered this book March 13. It is now May 21 and I still have not received my book. I'm not impressed. It was shown to be available when I ordered it.

Excitig!
I thought that unless one is a mathematician one cannot get excited about mathematics. I was wrong. I got a new appreciation for math and the book reads like a detective novel, history and adventure all in one. Fascinating. I gobbled it down in just a few days.

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Product Description
This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book. ... Read more

Customer Reviews (3)

Old school algebraic number theory with heavy Kummer bias
Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

great book
This is a great book.If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want.it really has a lot of stuff in it.all other graduate books are theory without examples or motivation.this book is the exact opposite.the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

Read this if you're seriously interested in math.
There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book. ... Read more

Product Description
This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University.Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable resource for mastering the epoch-making proof of Fermat's Last Theorem. ... Read more

Customer Reviews (3)

Yet another application of elliptic curves...
The successful proof of Fermat's Last Theorem by Andrew Wiles was probably the most widely publicized mathematical result of the 20th century. And once again, among their numerous other applications, elliptic curves are employed in the proof. The book is a compilation of articles written by first-class mathematicians, the reading of which will give one a thorough understanding of the proof, along with an overview of some very interesting topics in number theory and algebraic geometry. The reader who undertakes an understanding of the proof already no doubt has a substantial amount of 'mathematical maturity', and no review, no matter how complete, would influence greatly such a reader. Suffice it to say then that this book is excellent, and even a reader interested solely in elliptic curves and modular forms could benefit greatly from the reading of this book.

Great?!?!
This book might be good if you like number theory. But if you're an analyst who hates number theory or a brick-layer, then this book is probably not meant for you. I hope you found this review helpful. Have a nice day.

Highly recommended
This item is very instructively, not only for "real"mathematicians. Of course, sometimes it's very difficult to"read". It gives me pleasure to own the proof of FLT. ... Read more

Product Description
This new, completely revised edition of a classic text introduces all elements necessary for understanding The Proof (Title of a PBS series dedicated to the proof of Fermat's Last Theorem) as well as new development and unsolved problems. Written by two distinguished mathematicians, Ian Stewart and David Tall, this book weaves together the historical development of the subject with a presentation of mathematical techniques. The result is a solid introduction to one of the most active research areas of mathematics for serious math buffs and a textbook accessible to undergraduates. ... Read more

Customer Reviews (7)

Not bad, but much to be improved.
I entirely agree with the review by Mr T. Luo.In the parts I and II, there exist many logical gaps in the exposition that require a substantial amount of effort to fill in.If this book is used as a textbook in a class, that may prove pedagogically benefiting.But self-studying newcomers to the subject will find the textbook hard to follow.I must add that there are many typos concerning fraktur, especially in chapter 5, which makes the reading frustrating.

Great Introductory Book to Algebraic Number Theory
I wasn't lucky enough to have the opportunity to have a class in algebraic number theory in college or graduate school, so I had to learn it on my own. This book was recommended to me by my friend Paul Pollack (author of Not Always Buried Deep) and the suggestion was fantastic, as I was able to learn algebraic number theory.

The book is written very clearly, it has nice exercises that make the theorems clearer and it covers the basic concepts from algebraic number theory.

This a great book to learn the basics of the subject.

skips too much
I guess the previous reviewers didn't try any of the exercises in the book. They are very good problems but the text is far from sufficient for us to solve the problems. For example, there is only one example in chapter 2 on how to find integral basis and it is a quadratic field. However, the 4th problem of this chapter is to find the discriminant of a degree-4 extension! At least the author should supply more theorems on integral basis so that we know how to start such a problem.
I feel like the author is very "Rudin" in his writing, neglecting all the details. Sometimes it's fun to fill in the details myself, but sometimes it can be rather annoying. I think a undergraduate textbook shouldn't skip too many steps in the proofs.

tough problems => good for the student
The motivation of explaining Fermat's Last Theorem is a nice device by which Stewart takes you on a tour of algebraic number theory. Things like rings of integers, Abelian groups, Minkowski's Theorem and Kummer's Theorem arise fluidly and naturally out of the presentation.

The inclusion of problem sets in each chapter also enlivens its appeal to a student. Typically, the first problems in each set are easy. But later problems can be quite formidable, and really give a good mental workout of the salient issues just covered in the chapter.

Very clear introduction to Algebraic Number Theory
This book is a very clear intoductino to ANT.It is a good first step for many reasons.One: it stays with algebraic number fields that are extensions of Q, the rational numbers.You get a good feel for the subject.When you go to more advanced books Q is replaced by other fields (P-adic, function fields, finite fields,..).
Two: He assumes very little and writes very clearly
Three: You only needs to read his Galois theory book for the prerequisite
Four: His book is what is usually left for the reader to do as an excersize in more advanced books. ... Read more

Product Description
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. ... Read more

Customer Reviews (2)

An exciting book, but, beware!
This is an exciting book, but beware!

Hellegouarch claims to major in giving examples rather than proving "the basic structure theorems" in this book, and, this he does very well. The examples are beautiful. The proofs that he does offer are unusually elegant and instructive.This book is perfect for the serious student of mathematics who has had the usual undergraduate course covering things like the theory of rings and ideals, Galois theory, complex analysis, that sort of thing.

What really makes this book so perfect are the holes in the proofs. They are just exactly the right size to fill in with enough difficulty to strengthen your muscles but not to break your back. This book would be absolutely fantastic for a first or second year graduate course that used the Texas method to introduce the students to arithmetic geometry or albegraic number theory or modular forms, or any of a number of other sub-fields of mathematics.

A very strange thing, though, is that some of the holes in the proofs that are labelled "exercises" are trivial to fill in when compared with the real holes. I must qualify this statement by mentioning that I have only worked through the first 17 pages in detail. Although, I have read the whole book several times without paying too much attention to detail.

For example, on page 13, there is the statement, "since x and y are odd, p**2 + 3*q*2 must be odd." (x,y,p,q, are all integers, x=p+q, y=p-q, and GCD(x,y) = (x,y) = 1.) Now, using elementary number-theory odd-even type arguments, this is not obvious. However, computing modulo 2 makes it easy: p**2 + 3*q*2 == p+q = x == 1 modulo 2. Note also that the result does not depend on "y" being odd, as Hellegouarch's statement would have you believe. Figuring out mis-leading statements like this are a great way to prepare a young graduate student to become a research mathematician. In real research problems, you are not usually told which theorems to invoke to prove your results.

Two sentences later, he mentions that (p,q) = 1, which again requires a little thought. In the next sentence he applies a result (Corollary 1.6.1) proved for the Gaussian integers on the previous page, but, he claims that it is a Z[squareroot(-3)] form of this result that is really being used. Not so. It is the Gaussian integer [Z(i)] form that is being used. Furthermore, in applying Corollary 1.6.1, he uses not only the Corollary but side results that appear in the proof of the Corollary. Furthermore, he applies the Corollary in the highly special case when b=0 but doesn't tell you this.

For a professional PhD mathematician (like myself) figuring all this out was great fun, but, then, to further confuse the issue, when Hellegouarch gets to the bottom of the proof, he claims that the filling in of the final details are left to the reader as an "exercise." But, the final deatils are not an "exercise," they are immediately obvious, especially for the reader who has jumped the hurdles required to get to the end.

Another example of a hole which is a great exercise is the statement on the bottom of page 15 that if p is prime over the integers and reducible over the Gaussian integers, then the reduction is essentially unique. In other words, p can be written as x*y over the Gaussian integers where neither x nor y are units in essentially one and only one way. BTW, x is the conjugate of y in such a reduction. The proof follows easily by applying the norm function N(a+bi) = a**2 + b**2, but he doesn't tell you this. He doesn't even tell you that this is a hole that needs to be filled in. Noticing holes like this one are a great way for a young mathematician to be prepared for a career in research mathematics. Sometimes, such holes are not just little annoyances, but, real holes, and part of the work of a research mathematician is being able to find them. This was the case for Wiles first proof (1993) of Fermat's Last Theorem. It had a "real" whole and it tood a "real mathematician" to find it.

All this makes for great fun, but beware!

In working through most of the first 17 pages with a fine-toothed comb, I was struck by the lack of typos. I don't remember seeing any. Certainly not any that forced me to run a computation to decide whether or not it was a typo. But then, on page 18, I found the following statement,

"Euler introduced the ring Z[j] where j = exp(2pi*i/3) is a primitive root of unity, in order to study the Fermat equationn of degree 3; he accepted the fact that the fundamental theorem of arithmetic extends to Z[j] (fortunately for him this is actually the case, although it is not for the ring Z[squareroot(-3)])."

This statement sure looks false as it stands. There are various possibile explanation for it, but, the most likely is that "-3" is a typo and it should be "-5" or any other negative integer except for the nine integers that constitute H. M. Stark's 1967 solution to the "Gaussian number problem."

(The 9 values of "D" for which Q[sqr(D)] and hence Z[sqr(D)] are UFDs are -1, -2, -3, -7, -11, -19, -43, -67, and -163. Being a UFD is the usual interpertation of the phrase "the fundamental theorem of arithmetic extends to Z[sqr(D)]." Reference: Stewart and Tall (S&T) "ANT," 3rd edition, 2002, page 86. Although S&T do not call it "the Gaussian number problem, many other books do.)

Naturally, it would be nice if there were an errata sheet for a book like this that neither gives definitions of many of its terms nor gives proofs of its basic structure theorems (from which the definitions could be deduced). However, I could not find such a list on the web. John G. Aiken, PhD in 1972 in C* and W* algebras.

An excellent introduction
Modulo some sections that require more mathematical maturity, this book gives a straightforward introduction to the mathematics behind Fermat's Last Theorem that is accessible to the first or second year graduate student in mathematics. This is due not only to the excellence of the presentation, but also the many problems at the end of each chapter, making this book qualify more as a textbook than a monograph. Its perusal will give the reader an appreciation of the role of elliptic curves in the proof of Fermat's Last Theorem. Readers familiar with the applications of elliptic curves will find another impressive one in this context. It is a sizeable book filled with many definitions and theorems, so only a few features that make the book stand out will be mentioned.

The first of these is the chapter on elliptic curves, which the author keeps at a level that does not presuppose a heavy background in algebraic geometry. Instead, he develops them using an approach that one might find in elementary analytic or projective geometry. Mathematical rigor however is not sacrificed, and the author does not hesitate to use diagrams when appropriate. Readers therefore will find the presentation fairly easy to follow, and will not be stymied by the complicated constructions that can easily accompany discussions on elliptic curves in the context of Fermat's Last Theorem. The necessary algebra, such as Galois theory, is given in another chapter.

There are two "million-dollar" problems mentioned in this book, such as the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture. The Riemann hypothesis arises in the discussion of zeta functions for elliptic curves. In this context, the author characterizes the zeta function in a way that makes its role in number theory very transparent, namely in the role it plays for expressing an integer as a product of primes, and the fact that it can be associated with the valuations of non-zero ideals in the integers. Groups that are "simpler" than the integers, such as the p-adic integers, also have zeta functions and similar product representations. The need for zeta functions in the book comes in the context of elliptic curves E over the rational numbers Q. The fields "simpler" than Q are the finite fields F[p] modulo a prime p also result in a representation of the zeta functions as a product, but now the product is taken over the prime ideals of quadratic extensions of the polynomial ring F[p;X] generated by an elliptic curve over F[p]. By quoting, but not proving the Artin representation of the zeta function for E, the author uses this to motivate the `L-function' for E. The Birch-Swinnerton-Dyer conjecture comes in when considering the Mordell-Weil group of E, and asserts that the rank of this group is equal to the order of the zero of the L-function at 1.

In the very last section of the book, the author discusses some new areas and concepts in mathematics that were generated by the solution of Fermat's last theorem. One of these concerns a new definition of the ring of p-adic integers, and arises when considering the reduction of an elliptic curve modulo a prime number. For p = 3 or 5, showing that the impossibility of the case of Fermat's theorem for these values of the exponent must be done by the considering, not the congruence modulo p, but the congruence module p^2. The same holds for p = 7, where no h-th power of p will give the result modulo p^h. The author therefore considers infinite powers of p, which brings in the notion of a `projective limit.' Infinite products of the integers modulo prime powers, taken with the Tychonoff topology, gives a local ring on which one can define a p-adic valuation. The author then considers the fraction field of this ring, which is locally compact under the p-adic distance, is the completion of the rational numbers under the p-adic distance, and is isomorphic to the field of p-adic numbers.

The author then generalizes this construction by starting with an elliptic curve E over a field K, and for a prime number not equal to the characteristic L of K, he shows how to construct the `Tate module' T(E;L) of E at L. Taking projective limits in this case shows that T(E; L) is a free Z(L)-module of rank 2. For the Galois group G of the algebraic closure of K, the Tate module is also shown to be a G-module over Z(L). Given a prime number p, the Tate module T(E; L) allows one to do arithmetic just as easily, or just as hard, as one does arithmetic in a finite field F[L], if one views the arithmetic in the context of an elliptic curve over Q (one is thus justified in setting L = p). The elliptic curve and the Tate module allow one to know just how many points are in the reduced elliptic curve E in F[p], this following from an understanding of the representations of the Galois group for a fixed L (these representations are related to each other, and thus serves to make the prime arithmetic more manageable). This line of thought is continued by putting the loxodromic parametrization of elliptic curves into this context, resulting in "Tate curves" E[q] for a p-adic number q.The author ends this section by discussing briefly some conjectures that he feels will be major unsolved problems in the years. One of these, called `Szpiro's Conjecture', postulates that the minimal discriminant of an elliptic curve over Q is bounded by its conductor. The other, called the `abc Conjecture' conjectures that the maximum of the valuations of three relatively prime integers is bounded by the radical of the product of these integers. Consequences of these conjectures are briefly discussed, including an interesting generalization of Fermat's equation.

A very helpful historical summary of the "elliptic curve approach" to Fermat's Last Theorem is given in the appendix. ... Read more

Product Description
June 23, 1993. A Princeton mathematician announces that he has unlocked, after thousands of unsuccessful attempts by others, the greatest mathematical riddle in the world. Dr. Wiles demonstrates to a group of stunned mathematicians that he has provided the proof of Fermat's Last Theorem (the equation x" + y" = z", where n is an integer greater than 2, has no solution in positive numbers), a problem that has confounded scholars for over 350 years.

Here in this brilliant new book, Marilyn vos Savant, the person with the highest recorded IQ in the world explains the mathematical underpinnings of Wiles's solution, discusses the history of Fermat's Last Theorem and other great math problems, and provides colorful stories of the great thinkers and amateurs who attempted to solve Fermat's puzzle.
... Read more

Customer Reviews (35)

Another error she made
I stumbled upon this book while browsing in a bookstore long ago. I imagined from the title that she was attempting to explain to the public Sir Andrew Wiles' great accomplishment in proving Fermat's Last Theorem, but when I started reading her book, I couldn't believe my eyes.

In addition to the errors other reviewers have pointed out, she claims that non-Euclidean geometry is "false." I would send her my book on that subject but I doubt that she would read it.

I say that because a mathematician I know sent her a polite letter pointing out her errors. She replied:

"My mathematical friends and I had a good laugh over your missive. Keep those cards and letters coming."

Yes, she was smart enough to give the correct solution to the Monty Hall problem, but when she errs, she is either too arrogant or too ignorant or too corrupt to admit it.

Blame the publisher for Marilyn's mistakes
I picked this up cheaply at a used book shop based on Marilyn Vos Savant's reputation, she has the world's highest I.Q., and I was curious as to what her writing was like. It's quite good, as you would expect from a professional magazine writer. The subject was topical at the time (1993) and the book was quickly published to cash in on the announcement of a proof of Fermat's Last Theorem. Unfortunately, no one with college math was around to correct some repeated blunders. Just to pick one: Vos Savant confuses inductive knowledge, which she understands well, with inductive proof, which she doesn't.

Inductive knowledge is when we think we "know" something because we've seen it time and again. She illustrates it quite well with the phrase "all wheels are round". Why do we believe this? Because every time we see a wheel, it's round. Repeated exposure. Now to push things a little further, if "all wheels are round" then "something that's not round isn't a wheel". So logically every time we see a knife, which is not round, then this inductively confirms the earlier statement. Vos Savant's illustration drives the point home that this isn't really satisfying logically and that inductive knowledge doesn't really work when it comes to proofs.

What Vos Savant fails to realize is that an inductive proof is a completely different animal. It's an accident that both concepts use the same label (i.e. inductive). Inductive proof is a mechanical process, an algorithm, a method, by which we can show the truth of many mathematical theorems. To use the method, the theorem we're trying to prove must be about something we can count. We look at the first case and usually, it's fairly simple to show it to be true. Then we look at the second case, and again we often easily prove it to be true. Then we look at the third, fourth, fifth cases and we notice that we can generalize the next case from the one we are looking at now. So we do just that, we generalize: if we can show that the (n+1)th case (i.e. the next case) is true whenever the nth case (i.e. the case now) is true, AND if we explicitly show the first case to be true, then and only then is our theorem always true.

There is nothing weird or special about this, it only takes a little time and effort to understand. Proof by induction is not usually taught before college, and then only to those who take calculus, and then only superficially (analysis and abstract algebra courses examine this method (one of Peano's natural number axioms actually) in more depth). There's no denying that Ms. Vos Savant is an accomplished intelligent human being. But she only has, as we all only have, so much time alloted to her to enjoy life and pursue her interests. Mathematics is obviously not among these and shame on the publisher for thinking that Vos Savant's high I.Q. automatically made her a mathematical authority.

Vincent Poirier, Tokyo

A review of these reviews
Ah, these poor little threatened egos on display:How dare anyone contradict the university professionals?How dare anyone have an opinion outside the status quo?How dare any publisher even consider a book not from a fully affiliated tenured and chaired professor of mathematics?Where is peer review when we need it?I especially enjoyed the brag of vandalism.Savonarola would be proud.

Im SO excited
Isn't it just marvelous, finally a chance to polish up my knowledge of the worlds most famous maths problem. I can't tell you how much me and my wife have enjoyed, and been entregued by this. Night-after-night, just laying down on the hay, having a laugh with this top quality piece of material. Maths world here I come.

the world's IQest woman?
I have no problem to admit that the world's IQest person, is a female person. But as I browsed this book, I would be hard pressed to grant Madame Marilyn Vos Savant the title,as the world's IQest woman. ... Read more

Product Description
Around 1637, the French jurist Pierre de Fermat scribbled in the margin of his copy of the book 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 inquiry which culminated only recently with the proof of the theorem by Andrew Wiles.

This book offers the first serious treatment of Fermat's Last Theorem since Wiles's proof. It is based on a series of lectures given by the author to celebrate Wiles's 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 theorem as well as a brief discussion of Wiles's proof and its implications. Requiring little more than one year of university mathematics and some interest in 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 lesser-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.

Entertaining, controversial, even outrageous, this book not only tells us why, in all likelihood, Fermat did not have the proof for his last theorem, it also takes us through historical attempts to crack the theorem, the prizes that were offered along the way, and the consequent motivation for the development of other areas of mathematics. Notes on Fermat's Last Theorem is invaluable for students of mathematics, and of real interest to those in the physical sciences, engineering, and computer sciences-indeed for anyone who craves a glimpse at this fascinating piece of mathematical history.

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

This book displays the unique talents of author Alf van der Poorten in mathematical exposition for mathematicians. Here, mathematics' most famous question 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 number theory. 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 and intriguing collection of tidbits, anecdotes, footnotes, exercises, references, illustrations, and more.

Proving that mathematics can make for lively reading as well as intriguing thought, this thoroughly accessible treatment

Helps students and professionals develop a background in number theory and provides introductions to the various fields of theory that are touched upon
* Offers insight into the exciting world of mathematical research
* Covers a number of areas appropriate for classroom use
* Assumes only one year of university mathematics background 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 ... Read more

Customer Reviews (9)

One of the best books I have ever read
Look: I am a professor of mathematics (retired) and I did not understand much of the technicalities in this book. So the criticisms by other reviewers that the book was too difficult are correct. It is not an account for the lay person, although if read correctly, a lay person can still get a lot out of browsing through it, because the technical mathematics is peppered with so much history and anecdotes about the living mathematicians involved, plus the marvelous humor of the author, that a lay person could enjoy it greatly with the proper attitude.

There are other books about FLT written specifically for non-mathematicians. They pretend to explain what Wiles did, but that is practically impossible to do in any meaningful way. If you want to go that route, try "Fearless Symmetry" by Ash and Gross. If you think you understand the last chapters in it, maybe you are mathematically talented after all!

I rave about this book because, in addition to some technical information I am trained enough to understand, the author conveyed the incredible drama of Wiles' achievement so well, especially in contrast to the failures over hundreds of years of many expert mathematicians (well, they did have partial successes - Kummer, Vandiver, Faltings et al - but those were not front page news for the New York Times). And what added to that drama was Wiles' initial failure, an irredeemable gap in his first proof; fortunately he found a different method leading to a correct proof with the help of Taylor. Also, having read Wiles' final correct paper in the Annals of Mathematics (and not understanding it), I saw that he generously acknowledged all the preceding results and techniques by so many other fine mathematicians upon which his work was based. In other words, although Wiles deserves all the acclaim he has received, it still was primarily an achievement over a long period of time of a whole group of very able mathematicians who have not been knighted like Wiles. He is more like the player who wins the most valuable player award for the team that wins the Super Bowl or the World Series - it was the whole team that won, not just that player.

I highly recommend this book - yes, especially for those trained in mathematics, but also for those willing to read it just for the human interest aspects supplementing all that math., a great story told brilliantly.

good as motivation for a grad student
The following claim is way off the mark: "Assumes only one year of university mathematics background even for the more advanced topics."

The text will be usefull to graduate students who want to know what motivates the ideas used in the proof.As such the book is a usefull addition to the literature.

Neither recreational nor instructive
I quite agree with the reviewer from Massachusetts.
I bought this book in the hope that I could get enough (indices to the) information necessary to understand Wiles' proof of FLT contirbuted to Annals of Mathematics some ten years ago.
The book has simply turned out to be junk for me: it does not provide any enlightenment as to the undestanding of the proof, nor does it offer any recreational delight (supposed? by Poorten himself.) As many reviewers have pointed out, "arrogance" is the exact word to describe the attitude of the authour.
I too would like to have the money re-imbursed.
The bottom line is, if you would like to understand the proof, do not buy this book but follow the "beaten path": study algebra, algebraic number theory, class field theory, modular forms and elliptic curves. I know this sounds (and is) demanding, but it is not impossible since many good textbooks on each subject have appeared these ten years.

Not for the faint-hearted
Although the author comes over as arrogant,I am, after several years, warming to this book. With concentration and very careful reading I have found that much can be gained from it. It is humorous, witty and iconoclastic. Reading a page here and a paragraph there,I have learned what Mordell's theorem is,almost understood a single paragraph proof of the prime number theorem,and more maths besides.

It is however heavy going and the lectures and notes are in a concentrated form.The cover says the book assumes only one year of university maths.I really doubt this is enough math for this book.So beware.

Assumes Far More Than High School Math
This is grossly inaccurately advertised.In the introduction the author states that high school math plus an acquaintance with a first course in linear algebra is sufficient to understand the general flow. This is silly at best.

The contents are loosely related lectures introducing (and only introducing - this isn't a summary of Wiles' proof) topics in number theory necessary for proving FLT.Each lecture is followed by "Notes and Remarks" often containing more advanced material that is lengthier than the lecture itself.While this separation is good in itself, the lectures still require math far beyond high school and in some cases require graduate work.Lecture 4 starts with a cyclotomic field that is a concept well beyond high school.Lecture 8 starts with the Riemann zeta function that, despite the fact that a high school student can understand it as an infinite series, requires for its appreciation a mathematical sophistication that is not reached until graduate school.Lecture 12 contains the phrase "As regards the zeta function, the trick turns out to be to notice that ... is in fact holomorphic", so one must understand "holomorphic".Note 3 of lecture 13 refers to a residue that, as a topic in complex analysis, is unheard of in high school.Algebraic number fields, the Riemann sphere, poles of complex functions and more all make their appearance, albeit briefly.I truly picked these examples just by opening the book at random multiple times.Woe to the reader who is lacking these topics and more besides.

Pleasure to the reader with the background and, far more importantly, the mathematical sophistication to appreciate this book.As a set of lectures its character is quite different from a number theory textbook.Its audience is small but will no doubt be enthusiastic. ... Read more

Product Description
Proceedings of a conference at the Chinese University of Hong Kong, held in response to Andrew Wile's conjecture that every elliptic curve over Q is modular. The survey article describing Wile's work is included as the first article in the present edition. ... Read more

Product Description
The most significant recent development in number theory is the work of Andrew Wiles on modular elliptic curves. Besides implying Fermat's Last Theorem, his work establishes a new reciprocity law. Reciprocity laws lie at the heart of number theory.

Wiles' work draws on many of the tools of modern number theory and the purpose of this volume is to introduce readers to some of this background material.

Based on a seminar held during 1993--1994 at the Fields Institute for Research in Mathematical Sciences, this book contains articles on elliptic curves, modular forms and modular curves, Serre's conjectures, Ribet's theorem, deformations of Galois representations, Euler systems, and annihilators of Selmer groups. All of the authors are well known in their field and have made significant contributions to the general area of elliptic curves, Galois representations, and modular forms.

Features:

Brings together a unique collection of number theoretic tools.

Makes accessible the tools needed to understand one of the biggest breakthroughs in mathematics.

Provides numerous references for further study. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
Originally published in 1921.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

Product Description
This book concentrates on the final chapter of the story of perhaps the most famous mathematics problem of our time: Fermat's Last Theorem. The full story begins in 1637, with Pierre de Fermat's enigmatic marginal note in his copy of Diophantus's Arithmetica. It ends with the spectacular solution by Andrew Wiles some 350 years later. The Fermat Diary provides a record in pictures and words of the dramatic time from June 1993 to August 1995, including the period when Wiles completed the last stages of the proof and concluding with the mathematical world's celebration of Wiles' result at Boston University.

This diary takes us through the process of discovery as reported by those who worked on the great puzzle: Gerhard Frey who conjectured that Shimura-Taniyama implies Fermat; Ken Ribet who followed a difficult and speculative plan of attack suggested by Jean-Pierre Serre and established the statement by Frey; and Andrew Wiles who announced a proof of enough of the Shimura-Taniyama conjecture to settle Fermat's Last Theorem, only to announce months later that there was a gap in the proof. Finally, we are brought to the historic event on September 19, 1994, when Wiles, with the collaboration of Richard Taylor, dramatically closed the gap. The book follows the much-in-demand Wiles through his travels and lectures, finishing with the Instructional Conference on Number Theory and Arithmetic Geometry at Boston University.

There are many important names in the recent history of Fermat's Last Theorem. This book puts faces and personalities to those names. Mozzochi also uncovers the details of certain key pieces of the story. For instance, we learn in Frey's own words the story of his conjecture, about his informal discussion and later lecture at Oberwolfach and his letter containing the actual statement. We learn from Faltings about his crucial role in the weeks before Wiles made his final announcement. An appendix contains the Introduction of Wiles' Annals paper in which he describes the evolution of his solution and gives a broad overview of his methods. Shimura explains his position concerning the evolution of the Shimura-Taniyama conjecture. Mozzochi also conveys the atmosphere of the mathematical community---and the Princeton Mathematics Department in particular---during this important period in mathematics.

This eyewitness account and wonderful collection of photographs capture the marvel and unfolding drama of this great mathematical and human story. ... Read more

Customer Reviews (2)

The story of Fermat and the last theorem that took 300 years to solve
As a person who seriously studied abstract mathematics and one whose father was a number theorist before entering nuclear engineering, I have always had an interest in the difficult mathematical problems that today's mathematicians are tackling. It seems to me to be an achievement I never expected in my life time to see the four color problem and Fermat's last theorem both solved. My father died in 1971 and missed out on both. But I am sure he would have been particularly fascinated and interested in what Wiles and others did.
Given my interest and background, I still have to say that the subject of elliptic curves and modular forms is so abstract and specialized that I can't understand the language. The best I can make out of it is that there is this conjecture called the Taniyama-Shimura conjecture that is fairly old,dating back to the 1950s. Its proof implies Fermat's theorem even when specialized to a subset of the modular forms in the conjecture. Yet I know little about elliptic functions and almost nothing about modular forms.

In this little book Mozzochi, who understands what was going on, takes us through the events from the announcement in Cambridge by Wiles that he has attained results on Taniyama-Shimura that is enough to prove Fermat's Last Theorem as a corollary to the correction of the proof and finally a conference that studies and explains the results.

The book is just three chapters. Each represents a significant date. In Chapter 1 "February 10, 1994", Mozzochi discusses the dramatic presentation by Wiles in Cambridge and the events that followed and led to a discovery of a flaw in part of the proof. He also explains the results that led up to Wiles very secretive years of work on the problem.

Wiles is a bright mathematician who had an interest in mathematics at an early age but never seriously attempted to work on Fermat's problem when he embarked on his career as a mathematician. But his work was in the area of modular forms and elliptic functions and when Frey in 1985 proved that the Shimura - Taniyama conjecture implies Fermat's Last Theorem, Wiles knew he was at the right place at the right time to tackle this famous problem.

Wiles' introduction to his paper that appeared in the Annals of Mathemaics is contained in the book as Appendix A. The mathematical development that he describes is not comprehensible to anyone not specializing in this theory but it does show the logical thinking involved and the progression of results.

What is unique about this book is that Mozzochi was at many of the presentations that he describes in the book and he recorded them from an historical perspective through pictures, tape recordings, notes and email. Through email he presents some of the observations of other famous mathematicians who worked with Wiles to help correct the flaw in the proof. This includes Faltings and Taylor.

There are many nice pictures throughout the book showing the places and people involved in the story including a beautiful picture of Fine Hall in the fall of 1995(the Mathematics Department in Princeton where Wiles had his office and held his famous graduate lectures).

February 10, 1994 was the date that Wiles began a graduate class in Taplin Auditorium in Fine Hall where he would present the details of his results to the mathematics community and continue his quest to resolve the gap in the proof. It was at this time that his former student Richard Taylor joined him on his sabbatical from Cambridge in the effort. However, their work was kept quiet and they didn't even acknowledge to the public that Taylor came for that purpose, though many of the mathematicians there suspected it.

Chapter 2 titled "September 19, 1994" goes through the work that was done to close the gap. Wiles original approach was not working and he eventually had to abandon it but through the probing of Taylor he discovered why it was futile and how an earlier line of reasoning could work. This was like a revelation and Mozzochi expresses it best by quoting Wiles.

September 19, 1994 was the date of the revelation. The chapter explains the breakthrough that led to the final paper and the separate article with Taylor that patched the proof. In this chapter Mozzochi gives a nice description of Wiles as a mathematician and a truly "nice" person. He points out that although he was bright, Wiles was not considered the top mathematician in his field. That distinction belonged to Gerd Faltings a brilliant German who was a colleague of Wiles at Princeton. Faltings won the prestigious Fields medal and had revolutionized number theory. Some think that Wiles secretiveness was out of fear that Faltings would solve the problem first. However on April 21st 1995 Wiles delivered a lecture at Yale University where he detailed the proof. Faltings played an important role as the key referee who after reviewing the proof announced its correctness and in July 1995 presented a four page sketch of the proof in the Notices of the American Mathematical Society.

The final chapter "August 9, 1995" covers the conference that began on August 9th at Boston University. This instructional conference for graduate students in mathematics concentrated on Wiles' work and earlier related works.

There are as many pages of pictures as there are text. Mozzochi also presents a list of the famous mathematicians involved in the story as well as a listing of all the photographs. Appendix D with Ram Murty's review of the papers from the instructional conference at Boston University presents some detailed mathematics.

Buy this book if you are interested in Wiles' work from a historical perspective. It gives you a view of the man and the personal drama of trying to correct a proof that had been announced to the world and received tremendous media attention.

However, if you are looking for a deep understanding of the results look for a technical mathematics text. Any serious study of the mathematics would be difficult. As is pointed out by Mozzochi even at a one of Wiles' presentation for so-called laypersons, the room was filled with famous mathematicians including several Field's medalists.

The book achieves its goal. If you want a more detailed account, sketching the mathematical ideas for the layperson, the book by Singh is probably more appropriate.

Watching Wiles
Mozzochi's book is not a mathematics text, but a reporter's diary. He watches as Wiles' proof of the Modularity Conjecture for semi-stable elliptic curves unravels under the scrutiny of referees, and then as Wiles salvages his proof with the help of Taylor and others.

I expect that only people with a fascination for Mathematics will bother with this book - yet it is exciting to follow Wiles' struggles. Mathematics in the birthing has the same drama as ordinary life. There's a charming anecdote of a fellow who summarizes in a few words the encouragement and admiration of the public in chance encounters with Wiles.

There is little "character development", but Wiles' patience with the mathematical public during the vetting of his damaged proof is suggestive of his mettle. And his sense of humour complements his lectures well.

There is a history of the Modularity Conjecture, and an overview of the proof, linking Taniyama's conjecture to Fermat's, via Frey, Serre, Ribet, and finally Wiles. The list of references is excellent, as is Murty's review of the 95 Boston conference on Wiles' proof.

The pictures are valuable, but fewer (or smaller) would have been better. Blackboards aren't particularly charming.

A broader audience faces the high hurdle of abstruse mathematical concepts and language, but you're likely to enjoy yourself in spite of that. ... Read more