Amazon.com
When Andrew Wiles of Princeton University announced a solution of Fermat's last theorem in 1993, it electrified the world of mathematics. After a flaw was discovered in the proof, Wiles had to work for another year--he had already labored in solitude for seven years--to establish that he had solved the 350-year-old problem. Simon Singh's book is a lively, comprehensible explanation of Wiles's work and of the star-, trauma-, and wacko-studded history of Fermat's last theorem. Fermat's Enigma contains some problems that offer a taste of the math, but it also includes limericks to give a feeling for the goofy side of mathematicians.
Book 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.
Customer Reviews:
Great book.......2007-10-15
The book presents the history of number theory, starting from Pythagoras and culminating with Andrew Wiles, the mathematician who solved Fermat's Last Theorem. The author explains, for example, how and why negative and imaginary numbers were introduced, and sometimes throws in a riddle for the reader to solve (the answer is in the appendix). You don't need to know advanced mathematics in order to understand this book, since the author does a good job at explaining things in a way that everybody understands. Along the way, he gives an insight on the life of the great mathematicians, some of which are tragic. Good read, I highly recommend it!!!
One of the deep pleasures of my reading life..........2007-08-26
Once upon a time, two roads diverged into a yellow wood and I elected to pursue the path of an English and literature major. Along the neglected wayside lay a path toward mathematics, a discipline in which I was more academically proficient (not that I was bad at the other) but the purpose and pleasure of which had somehow escaped me. Though math had always been relatively easy for me, I believed -- perhaps through the way it had been taught and presented -- that it had never been "fun." Somehow its beauty had lain hidden beneath the drudgery of pedagogy. It was only years later that I discovered G. H. Hardy's "A Mathematician's Apology" and E. T. Bell's "Men of Mathematics" and learned to appreciate the joy of mathematics and its symmetry (ironically, the same appreciation that fires my love for Jane Austen). Though I will never become a professional mathematician, it is through books like "Fermat's Enigma" that I can share in the mathematician's joy of discovery that is the holy grail of every intellectual pursuit.
To ignore or to shy from this book because of a fear of math is to place unnecessary limits on your understanding of the world. Simon Singh does a masterful job of making accessible the story of an impenetrable math problem and telling it through the history of the mathematics that surround it and through the personalities of the mathematicians who pitted themselves against it and who ultimately led to its solution. There are no threatening equations here that are not solved for you -- only the challenge to get your mind around ideas that perhaps are new.
There is tremendous human drama along the way -- from the struggles of Sophie Germaine, who disguised herself as a man to study under the great mathematician Carl Gauss, to Evariste Galois, the brilliant young mathematician who died at the age of twenty in a dual he felt powerless to avoid, to Yutaka Tanyama, the brilliant Japanese mathematician whose conjecture pointed the way to Andrew Wiles' ultimate solution but who tragically took his own life before he was to ever see the proof, to the elation and doubts of Wiles himself when his much-heralded proof was found eventually to contain a fatal error. The story is not just a thumbnail history of the purest of sciences, but a story of human redemption as we see Wiles' struggles to correct his proof beneath the uncomfortable gaze of an anxious academic community.
Give yourself a chance on this one. I have read it time and again (and again) and have never failed to be moved by Wiles' ultimate triumph -- or in how articulately he uses metaphor to state his problem or the honesty with which he addresses his failures and shares his joy.
Informative, but not casual reading.......2007-08-23
Simon Singh makes difficult topics readable, although not necessarily totally understandable. This comment applies to Fermat's Enigma. The bulk of the book concerns the history of number theory and here it is truly outstanding. The description of the final solution was Jabberwocky to me.
Engaging even for the layman.......2007-06-11
Though some parts of it were way over my head (suffice it to say that math has never been my strong suit), it was still an interesting history of math and the seemingly unsolvable theorem that plagued it for hundreds of years. It was really fascinating to see how passionate some people are about math, how they have the sort of strong emotions for their subject that I traditionally ascribe to artists. This book did a really nice job of showing the astounding potential of the human mind and it offered some pretty astonishing history of the study of mathematics.
Simply Wonderful.......2007-04-16
There are two books that I have assigned my Algebra II class to read in the past - this was one of them (The other was "Bringing Down the House by Mezrich).
This book weaves together so many different themes :
The suspense and drama of Wiles solving the problem, wonderful diversions into mathematical history, and great insights into mathematical ways of thinking through interesting problems (like the mutilated chessboard, or the Bridges of Konigsburg problems)
I couldn't ask for a better book to teach my students what real mathematics is, in a fun and entertaining way.
Both professional mathematicians, and everday people who like math will enjoy this read.
Book 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.
Customer Reviews:
An exciting book, but, beware!.......2007-07-25
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.......2004-10-17
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.
Book 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.
Customer Reviews:
tough problems => good for the student.......2006-01-23
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.......2005-03-31
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.
thoughts from an amateur.......2004-02-12
good overview of algebraic number theory as it applies to FLT, however not exactly pitched at beginners. you'll want to have a grounding in abstract algebra & linear algebra at the minimum. still, even if you don't, you can get a good sense of the "big picture" and a high-level understanding of the advances in mathematics that were directly or indirectly related to attempts to solve FLT. overall a fascinating read if you're a math geek who wants something a little deeper than Simon Singh's pop treatment of Wiles' proof.
Lucid introduction.......2003-10-07
Lucid and clear introduction to algebraic number theory, in style very much like the author's other book on Galois theory. Very elementary though, doesn't cover any analytic method, nor gives even a taste of class field theory, besides the problem set is less than challenging. But the book serves its purpose well, strongly recommended for beginners.
Amazon.com
Born in 1601, Pierre de Fermat lived a quiet life as a civil servant in Toulouse, France. In his spare time, however, Fermat dabbled in mathematics, and somehow managed to become one of the great mathematical theorists of his century. Around 1637 he scribbled a marginal note in one of his books. In it, he stated that he had solved a celebrated number theory problem: "I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain."
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 mathematician named Andrew Wiles solved the riddle. Amir Aczel's account of this brainteaser and its solution is an irresistible read. And for mathematical dolts--like myself, for instance--it includes a concise, profusely illustrated history of mathematical theory from the Bronze Age to our own fin-de-siecle.
Book Description
Over three hundred years ago, a French scholar scribbled a simple theorem in the margin of a book. It would become the world's most baffling mathematical mystery.
Simple, elegant, and utterly impossible to prove, Fermat's Last Theorem captured the imaginations of amateur and professional mathematicians for over three centuries. For some it became a wonderful passion. For others it was an obsession that led to deceit, intrigue, or insanity. In a volume filled with the clues, red herrings, and suspense of a mystery novel, Dr. Amir Aczel reveals the previously untold story of the people, the history, and the cultures that lie behind this scientific triumph.
From formulas devised for the farmers of ancient Babylonia to the dramatic proof of Fermat's theorem in 1993, this extraordinary work takes us along on an exhilarating intellectual treasure hunt. Revealing the hidden mathematical order of the natural world in everything from stars to sunflowers, Fermat's Last Theorem brilliantly combines philosophy and hard science with investigative journalism. The result: a real-life detective story of the intellect, at once intriguing, thought-provoking, and impossible to put down.
Customer Reviews:
No mathematical depth and it sometimes reads like a children's novel.......2007-10-05
This is one of the books that appeared in print shortly after the announcement was made that Andrew Wiles had found a proof of Fermat's Last Theorem. In many ways, the public reaction to the announcement was surprising; there was a great deal of interest in the problem and therefore, publishers rushed to get a popular book out on the subject.
This book is one in the category of popular books, there is very little in the way of complicated mathematics and even then it is not really needed to understand the contents. Aczel weaves a complicated historical drama and often interjects verbiage more suited to a children's novel. For example, on page 133, there is the passage:
"Wiles walked around the department for several hours. He didn't know whether he was awake or dreaming. Every once in a while, he would return to his desk to see if his fantastic finding was still there - and it was. He went home. "
Many if not most of the greatest mathematicians of the ages are mentioned at some point in this book. While none of them are done in depth, there is enough for you to recognize the convoluted paths that solutions to complex problems often take. Furthermore mathematics is replete with "Aha!" moments where centuries of effort are suddenly distilled into a clear solution. Wiles had such a moment when he patched a serious hole in his original proof.
If you are interested in a detailed explanation of what Fermat's Last Theorem is and how it was proven, you need to look elsewhere. However, if a superficial explanation is the point of your interest and can tolerate some occasional poetic license in the area of exaggeration, then this book will work for you.
A Nice Little Book!.......2007-10-02
If you are looking for a little book about the history of Fermat's Last Theorem and how it was finally proved in 1995 by Andrew Wiles, this is the book for you, especially if you are living in the States where it will cost you a mere $[...]!(it cost me nearly double because of shipment charges by Amazon!). However, if you are serious about understanding how Wiles really proved the theorem, don't even bother! When we were kids, we had a joke about the teacher who asks the laziest kid in 5th grade to recite the table of multiplication by 9, and the kid goes something like:"Sir, I don't know the words, but I know the tune: tatata-Ta, tatata-Ta"...Well after reading this book, you will get to know the "tune" of Wiles's proof, sort of, and here it is:
- If there were solutions in natural numbers to x^n+y^n=z^n, then the elliptic curve known as "Frey curve" is semi-stable but not modular.
- All semi-stable elliptic curves ARE modular.
Therefore, the assumption about the existence of natural solutions to the above equation must be wrong!
Notice that as it is stated, the proof could fit in the margin of Diophantus' Arithmetica!!But the real proof has taken Andrew Wiles seven years of hard work, and 200 pages of very sophisticated maths!
At the beginning of the book, Aczel assumes his reader to be a mathematically illiterate person: he goes into long (and repeated!)explanations about rational numbers, irrational numbers, prime numbers, something that anyone who has heard about FLT should know very well. Then, when the real stuff comes, he starts talking casually about Galois Representations and semi-stable elliptic curves, as if the reader suddenly became a mathematics MA holder, at least! I know that the book is intended for the general public, but anyone who needs to be given a definition of prime numbers needs definitely an instruction on all the topics that make up Wiles' proof. All we find are scanty explanations, at best. So the reader who needs to understand should go for something else, and I suggest he or she starts by visiting Charles Daney's Home Page on www.mbay.net/~cgd/flt/fltmain.htm. There one finds all one needs to know about the subject, and it is for free!
But the book brings some useful insight into the way Wiles built up his proof, and most importantly on the ethics(or lack of!) of the international mathematical community: persons like André Weil and Jean-Pierre Serre, who are two very famous mathematicians and Field Medalists, appear in a very unfavorable light, and it is really a shame!
There are also a few pages needing revision for any possible second edition:
- At page6, there is a reproduction of Diophantus Arithmetica, with the following comment: "Pierre de Fermat's Last Theorem as reproduced in an edition of Diophantus' Arithmetica published by his son Samuel..." In fact, this is only the page of "Arithmetica" in the margin of which Fermat wrote his famous remark about the "marvelous proof" he had for his conjecture. No FLT appears anywhere in the page: it is merely a Latin translation of the original Greek on the subject of dividing a square into a sum of two squares...
- At page 92, Aczel writes: " Oddly, elliptic curves are neither ellipses nor elliptic functions". Then follows an explanation about elliptic curves, rational points on those curves, and modular forms, which is fine. But all of a sudden, at page 93, Aczel starts talking about "elliptic functions" as if they were elliptic curves.
- At the end of the book, Aczel asks what I find to be a rather facetious question on whether Fermat really had a proof, and thinks this might have been the case, or that the answer will never be known. I personally believe that, had there been a proof based on 17th century maths, Euler would have found it in less time than it took me to read Aczel's book! And if it had escaped Euler, Gauss would have no doubt found it(Gauss,arguably the greatest mathematical genius of all time, knew that he had no chance, so he did not even bother working on it!). The fact that FLT was only solved at the end of the 20th century definitely proves that it was unsolvable using anything but 20th century maths!
350 Year Old Detective Story.......2006-08-13
In 1637 Pierre de Fermat scribbled some notes in a margin of a mathematics book and the world has been talking about it ever since. Was his "last theorem" ever real or did he simply invent a story to boost his ego? Perhaps we will never know, but three centuries later Prof. Andrew Wiles did provide the solution to the greatest mathematical conundrum in history (or at least the most famous). Some people argue that Prof. Wiles used a mathematics unknown to Fermat, but if the Egyptians could build the pyramids in a mere 23 years (a feat that would take us with all our modern technology more than a century, if the money and determination could even be acquired) perhaps the knowledge of Fermat was lost over the centuries as well. We are certainly rediscovering things ancient people knew, so who is to say Fermat did not have a solution?
Though slow at times this is a fantastic detective story for anyone who loves mathematics (don't worry, you don't have to be a genius to enjoy it). A wonderful afternoon read.
An Introductory Book.......2006-05-04
While I did enjoy this book, I believe that it only serves as an introductory preface to the theorem. It does present all of the main ideas, but it fails to provide a network with which to tie them all together. When used with another book, such as Fermats Enigma by Simon Singh, many of the ideas come together so that the reader can see where Andrew Wiles proof came from and how it works. I would recommend this book for those who have very little knowledge of the theorem and would like to know a general outline.
Interesting idea, wrong direction.......2005-12-08
I think a book about the solution to Fermat's last theorem is an excellent idea. However, I think that this book ultimately fails because Aczel takes too much time discussing the history of the theorem and the developement of mathematics up until the point when the problem was solved in 1993. More than two centuries of mathematics is pigeon-holed into a book that is under 200 pages. In that respect, I think that the author was overly ambitious in his aim; it is simply impossible to explain topology in 3 or 4 pages. I would have rather seen the story approached from the human, rather than the math, side of things.
Book 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.
Customer Reviews:
Yet another application of elliptic curves..........2001-10-28
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?!?!.......2001-08-15
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.......2000-09-01
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.
Average customer rating:
- good as motivation for a grad student
- Neither recreational nor instructive
- Not for the faint-hearted
- Assumes Far More Than High School Math
- An illuminating and fascinating introduction to FLT
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Notes on Fermat's Last Theorem
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ASIN: 0471062618 |
Book 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
Customer Reviews:
good as motivation for a grad student.......2007-03-06
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.......2004-06-22
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.......2002-01-01
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. The book makes me want to take up number theory. Anyone out there want to finance me through further university studies?
Assumes Far More Than High School Math.......2001-03-21
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.
An illuminating and fascinating introduction to FLT.......1999-12-04
This is a splendid book. Covering advanced material yet remaining accessible to beginners is a difficult task, and van der Poorten succeeds admirably. Not only that, it's humorous, light-hearted, and in general a pleasure to read. I highly recommend it to anybody who wants more than a popular treatment of FLT yet doesn't have the background in algebraic number theory and algebraic geometry necessary to comprehend more advanced treatments. Come to think of it, I recommend it to anybody interested in mathematics at all. If you approach it with the understanding that all simplifications of advanced technical topics require accepting a bit of arm waving, I can pretty much guarentee that you'll love it.
Book 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.
Customer Reviews:
Old school algebraic number theory with heavy Kummer bias.......2006-01-10
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.......2002-08-08
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........2001-07-24
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.
Book 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.
Customer Reviews:
Difficult book but great topic coverage.......2002-09-04
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.......2002-09-03
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 !.......2001-07-02
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........1999-08-13
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 was able to follow the logic and math of all the presented proofs very well.
Customer Reviews:
This is a brilliant book; Hard to put down.......2006-10-09
It is hard to say enough good things about it. There are very few mathematicians that are interested in making math accesible to the general public and even fewer that are able to make it interesting and light to read. This author is one that is able to do both. (It's probably true that this book will only appeal to people who are analytical by nature, but have not gone through all the coursework that would intersect with this esoteric topic.)
If you have taken the basic high school sequence of: Geometry, Algebra II, Trigonometric Functions and Calculus, you can understand most of what is in this book.
The author does tend to digress a little into the stories of the lives of the mathematicians who made the math for Fermat's last theorem, but his providing of historical context was very interesting.
If you know nothing of Mathematics then you can still read this!.......2006-06-30
I'll make this short and sweet.
First off the book is an insight into the world of pure mathematics, it uses the fulcrum of the Last Theorem (an apparently very simple rule) to touch upon a number of different issues within both the practice and history of the subject.
Second. This book is interesting! It contains almost no technical material (aside from the optional appendices), and pure mathematics is surprisingly fascinating; not being a number theorist i appreciated the many references it has to the practical implications of maths, but there are also some noodle-baking moments concerning ideas that are so very simple, yet totally strange.
Third. Fairly quick to read.
Fourth. Tells the story of one of the most important breakthroughs of the twentieth century.
Hope this was useful
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