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33 of 36 people found the following review helpful
on May 15, 2005
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.
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32 of 35 people found the following review helpful
on May 25, 2003
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.
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17 of 19 people found the following review helpful
on June 18, 1999
This a great introduction to number theory, with a lot of the material directed to modern research. They discuss zeta functions, algebraic number theory, and elliptic curves. It is a helpful link from introductory number theory toward the vast fields of research in the area.
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13 of 16 people found the following review helpful
on December 20, 2003
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.
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20 of 26 people found the following review helpful
on December 6, 1999
If ever there was a textbook of which one could say that it was a thing of beauty, this has to be it. The book is very clearly written, and it is readily accessible even to those without a deep understanding of algebra or analysis; despite this, it manages to touch upon a great deal of relatively sophisticated material, and in a way that makes clear the links between the problems of the past and those of the present. I'd imagine that the book would constitute an essential item of reference for anyone with more than a passing interest in number theory.
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3 of 3 people found the following review helpful
on December 22, 2011
This book is a model of elegant and concise writing that is delightful to read ... provided you have the necessary background. By that, I mean a familiarity with (abstract) algebra at the undergraduate level, and a level of mathematical "maturity". The authors often provide proofs that are concise but clear. They demonstrate how, with a little algebra, we can acquire a deeper grasp of basic theorems like "Fermat's Little Theorem" (which is just something that drops out as a corollary once the appropriate lemmas and theorems are proved), and concepts like primitive roots, etc.

As far as coverage goes, it does not attempt a very comprehensive treatment of all the major topics in number theory. Thus, while multiplicative number theory is elegantly and insightfully treated, additive number theory is missing. Instead, the authors move from the foundations towards areas of current interest, such as elliptic curves. Perhaps that is why they call it "Modern Number Theory". The reader who wishes to study some of the more classical aspects of number theory could consult other texts like Hardy & Wright, or Niven.
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5 of 6 people found the following review helpful
on June 28, 2012
I am a self-studier so I thrive on texts that are self-contained, give beautiful proofs, and make eye-opening observations. (Not observations that add to perplexity.)

This book sets the gold standard for all of those criteria.

I bought this based on the glowing comments on two major, high-powered math forums - where the commentators were extremely well-versed in the material. They certainly knew what they were talking about.
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4 of 5 people found the following review helpful
on September 16, 2011
I have worked through all the problems in the first 8 chapters, and return to it constantly-have read most of the book one way and another. The (very readable) writing style really enables a student to understand an underlying theme of ideas well. A truly beautiful selection of topics that have helped in my own research in writing papers in number theory (especially the end notes of the chapters). A book that I regard with great affection, and will always carry with me. I can never completely express my gratitude to the authors sufficiently. It just occurred to me that rather than take my word for it, read the introduction to the book "Gauss and Jacobi sums" by Berndt, Evans, and Williams, in which Prof. B. Berndt, and Prof. R. Evans, both experts in number theory, explicitly credit this particular book as being their inspiration. It is one of the great number theory textbooks around today.
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on November 24, 2014
The book does require some background in algebra, but is a surprisingly easy read
considering the amount of material that it covers. Despite being 25 years old, it's a
good introduction to many topics in number theory.
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10 of 16 people found the following review helpful
on May 14, 2005
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
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