A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics)

by Kenneth Ireland (Author), Michael Rosen (Author)

16 customer reviews

ISBN-13: 978-1441930941

ISBN-10: 1441930949

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This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves.

A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics)

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Product Details

Series: Graduate Texts in Mathematics (Book 84)
Paperback: 394 pages
Publisher: Springer (December 5, 2010)
Language: English
ISBN-10: 1441930949
ISBN-13: 978-1441930941
Product Dimensions: 6 x 0.9 x 9 inches
Shipping Weight: 1.5 pounds (View shipping rates and policies)
Average Customer Review: 4.6 out of 5 stars See all reviews (16 customer reviews)
Amazon Best Sellers Rank: #2,027,001 in Books (See Top 100 in Books)
- #745 in Books > Science & Math > Mathematics > Pure Mathematics > Number Theory
- #17064 in Books > Textbooks > Science & Mathematics > Mathematics

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4.6 out of 5 stars

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Top Customer Reviews

ByA customeron 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.

0Comment 34 people found this helpful. Was this review helpful to you?YesNo

Best book on the subject

ByAdam Masseyon 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.

0Comment 38 people found this helpful. Was this review helpful to you?YesNo

ByA customeron 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.

0Comment 18 people found this helpful. Was this review helpful to you?YesNo

simply irresistible

Byapson June 28, 2012

Format: HardcoverVerified Purchase

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.

0Comment 7 people found this helpful. Was this review helpful to you?YesNo

A Modern Classic

ByAbiola O. Lapiteon 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.

0Comment 21 people found this helpful. Was this review helpful to you?YesNo

Covers many important areas

ByJulius Vernonon December 20, 2003

Format: HardcoverVerified 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.

0Comment 13 people found this helpful. Was this review helpful to you?YesNo

A model of elegance, and of concise writing

ByDaniel Wongon December 22, 2011

Format: PaperbackVerified 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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a fine piece of exposition

BySTUDENTon September 16, 2011

Format: PaperbackVerified Purchase

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.

0Comment 4 people found this helpful. Was this review helpful to you?YesNo

Most Recent Customer Reviews

excellent, a classic

Published 4 months ago by Roger HK Kadala

This book should not be sell in the US as ...

This book should not be sell in the US as it is against publication law. It is the Chinese reprint of the same American book, supposed to be sold only in China.Read more

Published 1 year ago by Peter

Very good book.

Published 1 year ago by Zhenchao Ge

but is a surprisingly easy read considering the amount of material that it covers

The book does require some background in algebra, but is a surprisingly easy read
considering the amount of material that it covers.Read more

Published 1 year ago by math student

Published on August 19, 2014 by noneyet

Excellent is was I expected in this product.

Excellent is was I expected in this product for my niece. She needed for study in the university. Ok, !!!!!

Published on March 20, 2013 by JL

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.Read more

Published on November 12, 2008 by Amazon Customer

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.Read more

Published on May 14, 2005 by Samuel Jacob Edwards

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