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A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) Hardcover – August 1, 1998

ISBN-13: 978-0387973296 ISBN-10: 038797329X Edition: 2nd

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Product Details

  • Series: Graduate Texts in Mathematics (Book 84)
  • Hardcover: 394 pages
  • Publisher: Springer; 2nd edition (August 1, 1998)
  • Language: English
  • ISBN-10: 038797329X
  • ISBN-13: 978-0387973296
  • Product Dimensions: 6.1 x 1 x 9.2 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.8 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #737,965 in Books (See Top 100 in Books)

Editorial Reviews

Review

From the reviews of the second edition:

K. Ireland and M. Rosen

A Classical Introduction to Modern Number Theory

"Many mathematicians of this generation have reached the frontiers of research without having a good sense of the history of their subject. In number theory this historical ignorance is being alleviated by a number of fine recent books. This work stands among them as a unique and valuable contribution."

— MATHEMATICAL REVIEWS

"This is a great book, one that does exactly what it proposes to do, and does it well. For me, this is the go-to book whenever a student wants to do an advanced independent study project in number theory. … for a student who wants to get started on the subject and has taken a basic course on elementary number theory and the standard abstract algebra course, this is perfect." (Fernando Q. Gouvêa, MathDL, January, 2006)


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Customer Reviews

4.8 out of 5 stars
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A book that I regard with great affection, and will always carry with me.
A. RAJAN
I have devoted a good portion of my life to the study of mathematics in general, especially algebra and number theory.
Julius Vernon
This book treats the topic quite nicely, making a fairly difficult concept accessible).
Adam Massey

Most Helpful Customer Reviews

32 of 35 people found the following review helpful By A Customer on May 25, 2003
Format: Hardcover
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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31 of 34 people found the following review helpful By Adam Massey on May 15, 2005
Format: Hardcover
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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17 of 19 people found the following review helpful By A Customer on June 18, 1999
Format: Hardcover
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 By Julius Vernon on December 20, 2003
Format: Hardcover Verified Purchase
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 By Abiola O. Lapite on December 6, 1999
Format: Hardcover
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 By Daniel Wong on December 22, 2011
Format: Paperback Verified Purchase
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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