This is not a book for learning number theory for the first time!
I have a B.S. in mathematics and I always did well in my courses; I was particularly good at number theory. My undergraduate class used Elementary Number Theory (5th Edition), which is actually a pretty great book. Looking for something more advanced, I signed up for an independent reading course, and this is the book the professor assigned.
First of all, I do not recommend this text unless you have a strong background in algebra. Number theory and abstract algebra are inextricably linked, and this book makes frequent use of the connections, but without doing much to explain anything that more solidly falls under the "algebra" heading. Without a good understanding of field theory, this book will be beyond your grasp.
This is, without a doubt, a "difficult" text. It's very terse, and while the proofs are elegant, they're often quite mysterious. I can't even count the number of times that the phrase "It's obvious that..." has left me completely mystified, and it's a gleeful moment when I can pencil in the margin that it actually IS obvious, for once. The exercises are frequently more difficult than it seems the author's intended; several of them have stumped my professor, and the motivation isn't always obvious.
This leads me to my main point: This is not a book for learning number theory for the first time! This isn't even a book for learning number theory for the second time. This is a book for developing an extremely rigorous understanding of a complex subject once you already have a wide variety of tools at your disposal and already possess a solid foundation in mathematics.
The difficulty level of the text isn't the reason for the "low" review score. The typesetting is, in several places, ambiguous. The notation can lead to confusion in even interpreting an exercise or statement. This seems to be mostly a result of lack of effort; I don't see a reason why the Legendre/Jacobi symbol can't always be made easily distinguishable from regular division. Context should help make the distinction, but if you're having a hard time understanding what's going on, the added level of frustration in simply interpreting the notation is just superfluously discouraging.
Essentially, this can be a challenging text to work through, and you'll find very little in the way of support in its pages. I've found myself turning to other references countless times to get a handle on some of the results, and I think a lot of that explanation could easily have been included in the first place. I'm not a fan of "elegant" math in the learning process; I'm a fan of explanations, examples, and connections... all of which are in extremely short supply in this text.
Best book on the subject
I am currently finishing my third year of undergraduate math at Brown University, and have just completed a course that used this particular book. I have to say it's the most WELL WRITTEN math book I've ever read, and I've read many, many math books by now (more than I'm willing to count as I'm typing this). Professor Rosen (and Ken Ireland, God rest his soul) have made a book that has both fun and interesting problems as well as clear explanations of proofs in the text. It does of course require that you know the basics of abstract algebra (in particular, one is expected to know that "1" is a unit and therefore cannot be prime, so of course when we discuss problems involving factorization into primes, one will of course ignore the number 1). One is also expected to know the basics of formal logic (i.e. understanding how a proof by induction works, how a proof by contradiction works, and knowing that any proper subset of the natural numbers will have a least element), and I choose to point this out simply because MrBigBeast's review makes it obvious that all these facts were not understood. Despite the fairly large amount of assumed knowledge (this is a book intended for advanced undergrads and first year grad students, afterall), this book takes one on an amazing adventure through the depths of elementary number theory, as well as introduces you to very advanced topics in both algebraic and analytic number theory (ever want to know about Zeta Functions? This book treats the topic quite nicely, making a fairly difficult concept accessible). Truly a gem of a book and worth buying even if you never use it for a course.
Great Book
I'm currently an undergrad math and phsyics major at Brown, and I loved this book. Rosen is a great teacher and a great writer. As per the post below mine, the submitter is being overly nitpicky. If a reader cannot realize that unique factorization of Z+ extends to Z or understand immediately the nature of "1", then perhaps the reader shouldn't be trying to learn advanced number thoery. As per using the conclusion in the proof, it's called proof by induction. It's easy and trivial enough that I'm sure they didn't want to waste the readers time going through the incredibly obviouse steps.
The book is great. The problems are fun and interesting, and the book gradually generalizes which makes the abstraction easier to conceptualize. If you need something with tons of really baisc excersizes and proofs that will walk you through every step of the way, no matter how small, then this book may not be for you. But if you are a seriouse student looking for an interesting and insightfull introduction to the subject, I highly recomend this book
Covers many important areas
I have devoted a good portion of my life to the study of mathematics in general, especially algebra and number theory. This book is an extraordinary reference to many areas of number theory and extremely approachable. The book can be studied on its own or as a companion piece to more specialized texts such as Marcus's Number Fields.
Simply Amazing
I picked up this book as a junior in college and was simply stunned. The flow of ideas is so natural that there are times when you can even read the book like a novel. The exposition is clean, and the proofs are elegant.
However, keep in mind that this book IS a GTM. Hence, it requires pre-requisites by way of approximately a year of abstract algebra. As the author says in the preface, it's possible to read a the first 11 chapters without it. However, to appreciate the beauty of the theory, I would sincerely recommend algebra as pre-req.
The first 12 chapters can be considered 'elementary' (not easy, just fundamental). The others are specialized algebraic topics. For instance, the chapter on elliptic curves is useful to get a flavor of the subject. However, it includes very few proofs.
(7 reviews)
