Customer Reviews

Algebraic Number Theory and Fermat's Last Theorem: Third Edition

5 star:		(2)
4 star:		(3)
3 star:		(2)
2 star:		(0)
1 star:		(0)

 

 

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.

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.

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.

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.

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.

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.

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.