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.
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Editorial Reviews
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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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Most Helpful Customer Reviews
23 of 24 people found the following review helpful:
5.0 out of 5 stars
Simply Amazing,
By A Customer
This review is from: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) (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.
22 of 25 people found the following review helpful:
5.0 out of 5 stars
Best book on the subject,
By Adam Massey "Adam" (Providence, RI USA) - See all my reviews
This review is from: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) (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.
16 of 18 people found the following review helpful:
5.0 out of 5 stars
Great Text!,
By A Customer
This review is from: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) (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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Inside This Book
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First Sentence:
As a first approximation, number theory may be defined as the study of the natural numbers 1, 2, 3, 4, . . . . Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
height regulator, global zeta functions, biquadratic reciprocity, cubic reciprocity, irregular primes, cubic character, power residue symbol, quadratic number fields, distinct prime ideals, canonical height, higher reciprocity laws, residue character, lth power, primary primes, projective hypersurface, ideal class group, multiplicative character, algebraic number field, many rational points, jacobi sum, prime decomposition, character modulo, descent argument, algebraic integers, primes congruent Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Fermat's Little Theorem, Hecke L-functions, Dirichlet L-functions, Graduate Texts, Disquisitiones Arithmeticae, Generalize Exercise, Applying Lemma, New Progress, Oeuvres Scientifiques New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
As a first approximation, number theory may be defined as the study of the natural numbers 1, 2, 3, 4, . . . . Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
height regulator, global zeta functions, biquadratic reciprocity, cubic reciprocity, irregular primes, cubic character, power residue symbol, quadratic number fields, distinct prime ideals, canonical height, higher reciprocity laws, residue character, lth power, primary primes, projective hypersurface, ideal class group, multiplicative character, algebraic number field, many rational points, jacobi sum, prime decomposition, character modulo, descent argument, algebraic integers, primes congruent Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Fermat's Little Theorem, Hecke L-functions, Dirichlet L-functions, Graduate Texts, Disquisitiones Arithmeticae, Generalize Exercise, Applying Lemma, New Progress, Oeuvres Scientifiques New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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100
books
cite this book:
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- Elliptic Curves: Diophantine Analysis (Grundlehren der mathematischen Wissenschaften) by S. Lang on 5 pages
- Number theory, Volume 20 (Pure and Applied Mathematics) by Author Unknown on page 338, Back Matter (1), and Back Matter (2)
- Elliptic Functions (Graduate Texts in Mathematics) by Serge Lang on page 364, Back Matter (1), and Back Matter (2)
- Mathematische Werke (AMS Chelsea Publishing) by Gotthold Eisenstein on page 225, Back Matter (1), and Back Matter (2)
- The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (v. 106) by Joseph H. Silverman on page 338, Back Matter (1), and Back Matter (2)
See all 100 books this book cites
- Arithmetic of Finite Fields (Chapman & Hall/CRC Pure and Applied Mathematics) by Charles Small in Front Matter, page 137, and Back Matter
- A Course in Computational Algebraic Number Theory (Graduate Texts in Mathematics) by Henri Cohen in Back Matter (1), and Back Matter (2)
- Introduction to Elliptic Curves and Modular Forms (Graduate Texts in Mathematics) by Neal I. Koblitz in Back Matter (1), and Back Matter (2)
- Number Theory in Function Fields by Michael Rosen in Back Matter (1), and Back Matter (2)
- Introduction to Cyclotomic Fields (Graduate Texts in Mathematics) by Lawrence C. Washington in Back Matter (1), and Back Matter (2)
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