Notes on Fermat's Last Theorem

Rating:
Summary: An illuminating and fascinating introduction to FLT
Review: 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.

Rating:
Summary: Assumes Far More Than High School Math
Review: 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. As someone who does understand most of it I attest that 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.

Rating:
Summary: Assumes Far More Than High School Math
Review: 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. As someone who does understand most of it I attest that 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.

Rating:
Summary: Not for the faint-hearted
Review: Unfortunately this book looks like it has been put together quickly to try to make a buck in the wake of Wiles' proof. The author is insufferably arrogant. In four decades of reading books I have never come across such arrogance in print. In the Introduction the author brags about the frequent flyer miles he collects jetting to math conferences. And we are boringly told about his interest in sport and science fiction. (I always think an interest in science fiction shows a mediocre mind.) Like Caesar, Mr Van der Poorten can follow sport, read science fiction and construct mathematical proofs--all at the same time!

The dust jacket says "Assumes only one year of university maths." Don't you believe it!

Having said all that, I did learn what Mordell's theorem was; and some other math as well. The book contains many useful references to other books.

The book is chatty, witty, informal and iconoclastic. Perhaps there are experienced number theorists who would appreciate it. But it is not for anyone else.