Product Description

This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.

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This book illustrates the application of Number Theory to practical problems in physics, digital information processing. computing, cryptography, acoustics, crystallography (quasicrystals), fractals and self-similarity. Its aim is to widen the horizon of readers with a minimum of mathematicasl training to the basic facts of number theory. The otpics treated informally, stressing intuition rather then formal proofs. The book demonstrates that there are a surprising numbe rof applications of a field which is tradiationally considered raterh abstract, and from this realization readers are led to a depp appreciation of tzhe usefulness of finite mathematics and its multi-faeted interactions with the real world. The Second Edition includes much new material on self-similarity, factals, quasicrystals, Cantor sets, Hausdorff dimensions, detemrinicstic chaos, errof-free computation, spread-spectrum communication systems, optimal ambiguity functions for radar and sonar, and Fibonacci numbers. From the reviews "A lighthearted and readable volume with a wide range of applications to which the author has been a püroductive contributor - useful mathematics given outside the formalities of theorem and proof. Philip Morrison - Scientific American NTS*** ... Read more

Customer Reviews (2)

Good mix of theory and mathematics
This book provides good examples and has a good mix of number theory and the associated mathematics. Very useful for people interested in cryptography and number theory in general.
However, this book is not easy to read and requires some effort to digest the given information.

could have included some problems
[A review of the 4th Edition 2006.]

It is possible when teaching number theory to drown the reader in theorems, lemmas and corrolaries. So much so that she can get lost in the thickets and fail to appreciate the broad motivating ideas. Schroeder refrains from such a presentation. He is certainly rigorous enough, when needed. But the book is a graceful exposition. Explaining key concepts and proving enough along the way to satisfy most readers.

So Euler, Fermat, Gauss and other luminaries make their appearance at numerous points. Along with the classic and still unproven Goldbach Conjecture. Many readers will probably turn to the sections on modern applications, notably in cryptography. The explanation of the public key algorithm is elegant.

Other applications include making random numbers. Something quite subtle and difficult to do well. And vitally necessary for cryptography.

The last chapter on fractals and self similar transformations is accompanied by a few pretty pictures of fractals and Julia sets in the plane. Though by now most readers must be familiar with fractal art.

The only drawback of the book is the lack of problems. Pity, as it reduces the book's suitability as an undergrad text. ... Read more

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Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that students can easily cross-reference and find the appropriate data.

With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an overview of Fermat’s Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue–Siegel–Roth theorem, Hall’s conjecture, the Erdös–Mollin-–Walsh conjecture, and the Granville–Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes’, Selberg’s, Linnik’s, and Bombieri’s sieves. He also discusses recent results on gaps between primes and the use of sieves in factoring.

By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at the graduate level.

Customer Reviews (1)

Number Theory applied !
The greatest 20th century Number Theorist Prof Hardy said "Number Theory" is useless. This book proves him wrong ! Today number theory is the most exciting field in Maths applied in computer, thru the later penetrates to every other applied science, economy, military, etc.

This book covers many interesting applications of number theory, although the title "advanced" sounds scary, but it is not difficult to read for maths undergrads.

... Read more

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"...a fine book ... treats algebraic number theory from the valuation-theoretic viewpoint. When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has seomething extra to offer. This is not surprising, for it was he who inaugurated the local-global principle (universally called the Hasse principle). This doctrine asserts that one should first study a problem in algebraic number theoy locally, that is, at the completion of a valuation. Then ask for a miracle: that global validity is equivalent to local validity. ... It is trite but true: Every number theorist should have this book on his or her shelf."- IRVING KAPLANSKY IN BULETIN _OF THE AMERICAN MATH SOCIETY ... Read more

Customer Reviews (1)

Number Theory
There has never been in any time a book in any language that can put on a par with Hasse's monumental refrence Number theory . It is a text as well as a great refrence, and it is sad that those people at springer verlag not just printed on a low quality papers, they put it altogether out of print. ... Read more

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Number theory shown to be virtually inexhaustible source of intriguing puzzle problems—interesting to beginning and advanced readers. Divisors, perfect numbers, the congruences of Gauss, scales of notation, the Pell equation, many other aspects produce ingenious puzzles. Solutions to all problems.
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Customer Reviews (8)

Full of info
...but poorly written.It's very interesting in a historical sense.For instance, Table 9 shows composite Mersenne's numbers up to n=251, most of them with question marks when a factor wasn't known to be composite or prime, and C when it was known to be composite but the factors weren't known.For instance, 2^101-1 was known to have at least 4 prime factors, but no factors were known.At the time (1964-1966), ENIAC was being used to compute such things.Yesterday, on my lowly Pentium 4, using Mathematica's built-in functions, I factored all the numbers completely in less than 45 minutes of computation time.One of them (2^251-1) took 34 minutes and 7 seconds and another took 5 minutes 10 seconds -- all the rest were very fast.It's amazing how much better the algorithms and computers are, today.

Full of info, but poorly written.
And NOT written in a recreational style, by any means.

Recreational Mathematics Is NotA Contradiction In Terms
If you are curious about numbers (as I am), this is one of several books you need to have in your library. It is informative, entertaining, comprehensive and easy to read, even if you are not blessed with a mathematics degree from an institution of higher learning.

This book is well-organized, beginning with topics that are easy to grasp, then going on to more complicated ideas. The contents include such terms as prime numbers, perfect numbers, amicable pairs, sociable numbers, divisors, congruences, cyclical numbers, repunits (a term the author coined) and logarithms. The book reveals many curiosities about common artcles such as squares, triangles and circles. More complex topics such as a resolution of a number into prime factors and Pell equalities are shown in a manner that is informative and easy to comprehend. The grand climax of the book is a stirring discussionof Fermat's Last Theorem.

This volume remains a 'good read' despite the fact that it is almost forty years old. Although much of what it in this book has been superceded over the course of time, this boof is not obsolete. If you have any interest in numbers, buy this book if you do not already possess a copy of it.

A classic
This is a great book for math amatuers like myself.Expertly written.Great introduction.Great examples.A must read for anyone interested in learning more about the pure science of number theory.

Lots of examples, lots of tables, lots of fun
This book was my first exposure to Number Theory, coming from an engineering background. This book got me hooked. You could almost say the book takes an engineering approach to number theory: Lots of examples, lots of tables but not a lot of rigorous, long proofs.This book is set apart from most textbooks by those facts. You won't find such extensive examples and tables in any textbook (and I have about 50+ texts on Number Theory). Conversly, no self-respesting professor could present so many results without proof. But that's the fun of this book. I recommend this book over"An Adventures Guide to Number Theory" by Richard Friedberg for uninitiated. ... Read more

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Customer Reviews (8)

An understandable journey through one of the most fascinating areas of human endeavor
Number theory is the most fascinating area of mathematics; I have yet to meet a mathematician that does not enjoy it. Making it even more interesting is the fact that the fundamental principles can be understood by almost everyone. It is the only area of mathematics where many of the most difficult unsolved problems can be understood by people having only a high school mathematics background.
This book is an introduction to number theory that will appeal to everyone with an interest in numbers that understands high school algebra. It begins with a history of the development of numbers and then quite correctly proceeds to a description of some of the most prominent number patterns. Congruence arithmetic, Diophantine equations, prime and lucky numbers, continued fractions and Fibonacci numbers are the remaining primary topics. Secondary topics are some mathematical curiosities and brief histories of some calculating prodigies.
If you enjoy numbers and are interested in learning more about their properties then this is a book that you will enjoy. High school algebra is the only prerequisite to this journey through one of the most fascinating areas of human endeavor.

Great service
When I purchase products directly from Amazon, I receive them fast. What I like more from Amazon is that it can combine the shipping to reduce my cost. I have been a customer of Amazon for more than 5 years. Amazon is a great service.

An Enjoyable Romp Through Number Theory, Skipping from Topic to Topic
Excursions in Number Theory (Oxford, publ. 1966; Dover reprint 1988) is a brief pleasure trip across the realm of number theory.C. Stanley Ogilvy's and John T. Anderson's enjoyable text only requires that readers have familiarity with algebra and have a penchant for puzzles. For those interested in more mathematics twenty pages of explanatory notes are found in the appendix.

Using carefully selected examples, the authors present key topics with surprisingly clarity.Although congruences (arithmetic, not geometric), Diophantine equations, and continued fractions may be unfamiliar, the reader rather quickly appreciates the critical roles played by these concepts and tools.For example, congruences prove to be exceedingly helpful in solving a wide range a numeric problems and also reappear in later discussions on irrationals, iterations, and Diophantine equations.

The study of prime numbers is fundamental to number theory, but as yet we have no known formula to produce all primes. Even more disturbing, we have no procedures that are even guaranteed to produce only primes (i.e., not yield an unpredictable mix of primes and composite numbers).There is something fundamental about primes that we seem not to understand.The short chapter, Prime Numbers as Leftover Scrap, offers a fascinating perspective that I have not encountered elsewhere.

Other chapters are more playful, offering curios, puzzles, and oddities.Some examples appear to be little more than amusing numeric coincidences while other oddities prove to have theoretical significance. I am not an avid fan of mathematical puzzles, but I thoroughly enjoyed these diversionary chapters.

As a follow-up to Ogilvy and Anderson, I am now reading:

Number Theory and Its History by Oystein Ore (1948), available as a Dover reprint (1988), is now rather old, even pre-dating computer use in number theory research. The difficulty level is moderate. The historical background is interesting. (3 stars)

Elementary Theory of Numbers (1962) by William J. LeVeque offers detailed proofs underlying number theory and should appeal to readers that enjoy studying mathematics. Topics include congruences, powers of an integer modulo m, continued fractions, Gaussian integers, and Diophantine equations. The Dover reprint suffers from a small font size. (3.5 stars)

Yet another Dover reprint, Number Theory (1971) by George Andrews also targets more serious readers. Andrews uses an interestingcombinatorial approach to number theory. Good font size and open page layouts. (4 stars)

Fantastic Journey
This is a reprint of one of the books that most inspired
my interest in mathematics as a boy. I highly recommend
it to any high school student interested in mathematics
or perhaps as a gift that might stimulate interest.
It is very short and very readable.I also recommend
the book "Excursions in Geometry" by Ogilvy.

A Wonderful Trip
Unlike other Dover books, this text does not require an extensive background in math and fluency in the language of proofs.It is, as the title suggests, a delightful excursion through number theory that will ignite your interest in the subject and move you to further study.

I found the author's annotations helpful and I did not mind the occasional use of British vernacular.At many points in the text, Ogilvy & Anderson prompt the reader to pursue a question on their own, rather than walk through a full proof or explanation.This may seem abrupt, but it keeps the text focused and leaves the reader wanting to know more about number theory.

I hope Dover continues to reach out to a general audience with books like this.It condenses a difficult subject into everyday language without condescending to the reader. ... Read more

Product Description
Classic two-part work now available in a single volume assumes no prior theoretical knowledge on reader's part and develops the subject fully. Volume I is a suitable first course text for advanced undergraduate and beginning graduate students. Volume II requires a much higher level of mathematical maturity, including a working knowledge of the theory of analytic functions. Contents range from chapters on binary quadratic forms to the Thue-Siegel-Roth Theorem and the Prime Number Theorem. Includes numerous problems and hints for their solutions. 1956 edition. Supplementary Reading. List of Symbols. Index.
... Read more

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Covers the main theorems of algebraic number theory, including function fields over finite constant fields. ... Read more

Customer Reviews (1)

A warning to beginners
Experts find this a very good book, and I rate it on their advice. But others need to understand that Weil is making a bit of a joke with the title. This book is "basic" in the sense that it proves the theorems that Weil feels organize and clarify all of number theory--the "basic" theorems in that sense. It is an introduction to class field theory.

As Weil says at the start of the book, it has few prerequisites in algebra or number theory, except that the reader is presumed familiar with the standard theorems on locally compact Abelian groups, and Pontryagin duality and Haar measures on those groups. This part is not a joke.

If you want to really understand class field theory this may be a good book. (I am reliably told it is.) But Weil deliberately avoids using many ideas that are now standard: geometric ideas such as group schemes, and especially cohomological methods.

Beginners studying algebraic numbers do not need this book. Weil recommends Hecke ALGEBRAIC NUMBERS for such readers, and that is a terrific book. To learn class field theory today you'd probably do better with and Cassels and Frohlich ALGEBRAIC NUMBER THEORY, which Weil also recommends in a note to the second edition of this book. ... Read more

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This new, completely revised edition of a classic text introduces all elements necessary for understanding The Proof (Title of a PBS series dedicated to the proof of Fermat's Last Theorem) as well as new development and unsolved problems. Written by two distinguished mathematicians, Ian Stewart and David Tall, this book weaves together the historical development of the subject with a presentation of mathematical techniques. The result is a solid introduction to one of the most active research areas of mathematics for serious math buffs and a textbook accessible to undergraduates. ... Read more

Customer Reviews (7)

Not bad, but much to be improved.
I entirely agree with the review by Mr T. Luo.In the parts I and II, there exist many logical gaps in the exposition that require a substantial amount of effort to fill in.If this book is used as a textbook in a class, that may prove pedagogically benefiting.But self-studying newcomers to the subject will find the textbook hard to follow.I must add that there are many typos concerning fraktur, especially in chapter 5, which makes the reading frustrating.

Great Introductory Book to Algebraic Number Theory
I wasn't lucky enough to have the opportunity to have a class in algebraic number theory in college or graduate school, so I had to learn it on my own. This book was recommended to me by my friend Paul Pollack (author of Not Always Buried Deep) and the suggestion was fantastic, as I was able to learn algebraic number theory.

The book is written very clearly, it has nice exercises that make the theorems clearer and it covers the basic concepts from algebraic number theory.

This a great book to learn the basics of the subject.

skips too much
I guess the previous reviewers didn't try any of the exercises in the book. They are very good problems but the text is far from sufficient for us to solve the problems. For example, there is only one example in chapter 2 on how to find integral basis and it is a quadratic field. However, the 4th problem of this chapter is to find the discriminant of a degree-4 extension! At least the author should supply more theorems on integral basis so that we know how to start such a problem.
I feel like the author is very "Rudin" in his writing, neglecting all the details. Sometimes it's fun to fill in the details myself, but sometimes it can be rather annoying. I think a undergraduate textbook shouldn't skip too many steps in the proofs.

tough problems => good for the student
The motivation of explaining Fermat's Last Theorem is a nice device by which Stewart takes you on a tour of algebraic number theory. Things like rings of integers, Abelian groups, Minkowski's Theorem and Kummer's Theorem arise fluidly and naturally out of the presentation.

The inclusion of problem sets in each chapter also enlivens its appeal to a student. Typically, the first problems in each set are easy. But later problems can be quite formidable, and really give a good mental workout of the salient issues just covered in the chapter.

Very clear introduction to Algebraic Number Theory
This book is a very clear intoductino to ANT.It is a good first step for many reasons.One: it stays with algebraic number fields that are extensions of Q, the rational numbers.You get a good feel for the subject.When you go to more advanced books Q is replaced by other fields (P-adic, function fields, finite fields,..).
Two: He assumes very little and writes very clearly
Three: You only needs to read his Galois theory book for the prerequisite
Four: His book is what is usually left for the reader to do as an excersize in more advanced books. ... Read more

Product Description
This book thoroughly examines the distribution of prime numbers in arithmetic progressions. It covers many classical results, including the Dirichlet theorem on the existence of prime numbers in arithmetical progressions, the theorem of Siegel, and functional equations of the L-functions and their consequences for the distribution of prime numbers. In addition, a simplified, improved version of the large sieve method is presented. The 3rd edition includes a large number of revisions and corrections as well as a new section with references to more recent work in the field. ... Read more

Customer Reviews (3)

If you want to be an analytic number theorist
Work through this book. While Serre's A Course in Arithmetic (Graduate Texts in Mathematics) is slicker, it is nowhere near as enlightening. Iwaniec's treatise Analytic Number Theory (Colloquium Publications, Vol. 53) (Colloquium Publications (Amer Mathematical Soc)) is a good reference for professionals, but unreadable for someone who has not seen (a lot of) the material before. Davenport's book is very clear and very deep at the same time. The recent editions of this book have been brought up to date, but the core has not changed too much, so don't feel obligated to buy the latest edition.

An extraordinary Book
Ever since I first read about the prime number theorem, I have been roaming the mathetmatical landscape, looking for the best proof of this result. I believe this book has it. It's not the simplest or the shortest proof, but it gives the deepest understanding of why the prime numbers behve like they do. In addition to this, it shows you the historical perspective in these proofs. All too often today math books give one short and slick proofs that leave you wondering how on earth they came up with it. In this book, however, one can almost feel the thoughts going through Riemann and Dirichlet's heads as they came up with the theorems. This book also has the proof of Dirichlet's theorem and Vinogradov's partial proof of the ternary goldbach conjecture. The vinogradov and following sections are considerably harder, partly because they were not written by Davenport himself. Anyway, if you're serious about Analytic number theory and how mathematicians think, this books needs to be on your bookshelf.

A good historical approach to Analytic Number Theory
I like this book because it gives you a good understanding of where the difficulties in the subject are. It takes a historical approach, following more or less the same steps that the original discoverers of these results took. Today we have very slick proofs for many of these results, and it is sometimes hard to understand why it took so long to discover them in the first place, but this book will give you this understanding; Dirichlet in particular practically had to invent Analytic Number Theory to prove his theorem on primes in an arithmetic progression.

The book works up gradually to each result, for example proving Dirichlet's theorem first for a prime modulus (as Dirichlet did himself), then the general modulus. In most cases it proves first the result for all primes (zeta function) and then the generalization for primes in an arithmetic progression (L function), pointing out which parts generalize easily and which cause special difficulties.

Some of the more advanced results covered are exponential sums, Vinogradov's theorem that every large odd number is the sum of three primes, and Bombieri's theorem about the average distribution of primes in arithmetic progressions.

I haven't seen the previous (1980) edition; this new edition seems to be lightly revised from the previous one. The last chapter is up-to-date and gives a brief survey of new results and of new books on the subject. ... Read more

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A Guide to Elementary Number Theory is a 140 pages exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams.

Though Plato did not quite say, He is unworthy of the name of man who does not know which integers are the sums of two squares he came close. This Guide can make everyone more worthy.

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An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
... Read more

Customer Reviews (2)

underdeveloped and outdated
This review refers to the 1965 Hardcover version of the book.

It's quite apparent that the 40 years that have passed since this book was printed have very much dated it's content. The definitions of many key concepts (such as an ideal) contain the right ideas, but are not formulated in the modern viewpoint.These, however are only minor setbacks. The main flaw of this book is its subject matter. There are 11 chapters, and it was not until the eighth that the ideas start getting deeper. Even these last 4 chapters do not delve very far into the heart of things.

The text is written with the reader in mind (almost excessively so). Useful equations are clearly labeled and the steps in the proof are clearly outlined, though sometimes to an unnecessary degree.

I would recommend this book for a mathematics hobbyist, or perhaps an undergraduate number theory course. For anyone with a stronger background, they wil not glean much.

A Strong Introduction
Proceeding from the Fundamental Theorem of Arithmetic, into Fermat'sTheory for Gaussian Primes, this book provides a very strong introductionfor the advanced undergraduate or beginning graduate student to algebraicnumber theory.The book also covers polynomials and symmetric functions,algebraic numbers, integral bases, ideals, congruences and norms, and theUFT. ... Read more

Product Description
This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students. ... Read more

Customer Reviews (2)

Really a treasure
I'm a student digging into the cryptology for an year. The more article I read, the more confusion I encounter because of my poor mathematical background. However, when I get this, I could find answer to my puzzles, and make an more explicit way to settle down my own idea.

The background you really need, clear and sweet
This book is a marvel.It is clear and concise yet thorough.The author is obviously a bit of an obsessive compulsive, he has found the shortest paths from the clearest definitions to the most important results, each given with the cleanest, most insight-inducing proofs ... the results (and definitions) he gives are the ones any student (practitioner!) of modern computer science (especially cryptology) *needs* to know -- having this book on your shelves (and its contents in your head) should be a requirement for any degree, at any level, in computer science.
[Caveat: I know the author and have read his book in draft form.I also required my students to get it and read it, in a computer science course I taught.] ... Read more

Product Description
First printed in 1967, this book has been essential reading for aspiring algebraic number theorists for more than forty years. It contains the lecture notes from an instructional conference held in Brighton in 1965, which was a milestone event that introduced class field theory as a standard tool of mathematics. There are landmark contributions from Serre and Tate. The book is a standard text for taught courses in algebraic number theory.This Second Edition includes a valuable list of errata compiled by mathematicians who have read and used the text over the years. ... Read more

Customer Reviews (2)

Hallelujah!Back in print at last!
This outstanding text has been out of print for over ten years, and during that time was the most wanted out-of-print number theory book.The travesty is ended and it is finally available to graduate students everywhere.Tate's thesis alone is worth the price of the book, but the reader is also treated to excellent expository articles on class field theory by top-notch mathematicians.

Now if we can just get Borevich and Shafarevich back in print too...

The best place to learn Class Field Theory
This book is one of the best places to learn class field theory. It is a true classic. ... Read more

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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.

Customer Reviews (7)

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 aspre-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. ... Read more

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This book contains discussions of hundreds of open questions in number theory, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In the second edition, extensive new material has been added, and corrections have been included throughout the book. This volume is an invaluable supplement to any course in number theory. ... Read more

Customer Reviews (2)

An excursion into the labyrinths carved by numbers
Number theory is the most enigmatic of disciplines, in that the problems are so easy to state and understand and yet so hard to resolve. Furthermore, when solved, the proofs are sometimes surprisingly easy. Inthis collection, Guy has put together a truly fascinating survey of what iscurrently (un)known about numbers.
Each page is an excursion into theextensive labyrinths carved out by numbers. Approximately once a month, Iscan it looking for new avenues to explore. Invariably, I see something,sketch out some possible proof routes and then end in frustration. Atypical result of working in number theory.
Whether you are anamateur or professional, if you have an interest in number theory, you willlike this book. Perhaps you will be able to make some progress towardsresolving some of these problems. It is certainly possible, as no field hashad more positive contributions from amateurs than number theory. EvenFermat fit the definition of an amateur.

Outstanding source of intersting problems in number theory
Another of my 10 favorite books.A constant source of inspiration ... Read more

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Eminent mathematician, teacher approaches algebraic number theory from historical standpoint. Demonstrates how concepts, definitions, theories have evolved during last 2 centuries. Abounds with numerical examples, over 200 problems, many concrete, specific theorems. Numerous graphs, tables.
... Read more

Customer Reviews (6)

All about quadratric number fields, but little else
From 1962, this is a detailed account of quadratic number fields, and makes a fair introduction to the theory of number fields of any degree. Ideal theory (restricted to the quadratic case) is well covered in plenty of detail. Gauss's classic theory of binary quadratic forms is also included.

Cohn is clearly quite keen on the subject, and is not just writing a textbook on some arbitrary topic for which he thinks there might be a market. And he has no fear of including pedagogical remarks in a textbook. The English is a bit awkward in places, but that is a minor thing.

The basics about characters and Dirichlet L-series are developed, but only to the extent needed to give Dirichlet's original proof of his theorem on arithmetic progressions. That proof, unlike later ones, uses Dirichlet's class number formula for quadratic fields, and is worth a look.

There is a lengthy but now dated bibliography.

An unusual feature is a table (from Sommer's 1911 book) describing the structure of Z[sqrt(n)] for all nonsquare n from -99 to 99.

Advanced, but now dated. Still useful.
This is definitely an advanced book. But no book claiming to be advanced can hold that title for long since mathematical research is progressive. As advanced as the book is, it'sjust an introduction to advanced number theory now, and dated in places.This book was orginally published as "A Second Course in Number Theory " in 1962. I own several books by Harvey Cohn and I appreciate his writing style. He writes with the complete book in mind and every chapter and paragraph is cohesively developed. His writing (between the numerous equations, tables and proofs) is lucid and conversational with historical motivation. He places a strong emphasis on ideal theory, and quadratic fields. In this regard the book is almost redundant given his "Class Field Theory" book.

Be warned there is some dated material in this book. It is prior to Alan Baker's 1966 proofabout d=-163 and imaginary quadratic fields, and such is still only conjectured in the text.And of course, FLT wasn't on Wiles' check list when this book was published.

It doesn't cover prime-producing polynomials or transcendental functions and their relation to class field theory, like one would hope (I guess the world had to wait for Baker for that). And forget about rational points on elliptic curves, none at all. It's from the period when elliptic equations were poo-pooed as relics before being brought to the fore again by recent developments.

Despite all the short-comings, I can still recommend the book as a worthy edition to your number theory library. Just don't put it at the top of your lists (unless you're short on cash and Dover is all you can afford).

Not advanced enough for Fermat's last theorm
For the money this book is a good buy. If you want to understand Wiles's use of modular forms in his proof of Fermat's last theorem, it isn't advanced enough! It also isn't an easy read, but it tries to cover themajor areas. It is best in quadratic number theory and worst in DirichletL-series and Gaussian Sums, but it mentions just about every area of numbertheory. If you want "easy" ... look elsewhere.

Not advanced enough for Fermat's last theorm
For the money this book is a good buy. If you want to understand Wiles's use of modular forms in his proof of Fermat's last theorm, it isn'tadvanced enough! It also isn't an easy read, but it tries to cover themajor areas. It is best in quadratic number theory and worst in DirchletL-series and Gaussian Sums, but it mentions just about every area of numbertheory. If you want "easy" as well as cheap, look elsewhere.

A book only for advanced students !
by the name itself you'll understand that the book is not for patzers, so don't take it for your introduction towards number theory..get a grasp of number theory from some other books and then polish your skills using thisone . ... Read more

Product Description

The book aims to take readers to a deeper understanding of the patterns of thought that have shaped the modern understanding of number theory. It begins with the fundamental theorem of arithmetic and shows how it echoes through much of number theory over the last two hundred years.

Product Description
A Course in Computational Number Theory uses the computer as a tool for motivation and explanation. The book is designed for the reader to quickly access a computer and begin doing personal experiments with the patterns of the integers. It presents and explains many of the fastest algorithms for working with integers. Traditional topics are covered, but the text also explores factoring algorithms, primality testing, the RSA public-key cryptosystem, and unusual applications such as check digit schemes and a computation of the energy that holds a salt crystal together. Advanced topics include continued fractions, Pell's equation, and the Gaussian primes.

The CD-ROM contains a Mathematica? package that has hundreds of functions that show step-by-step operation of famous algorithms. (The user must have Mathematica in order to use this package.) Also included is an auxiliary package that contains a database of all 53,000 integers below 10^16 that are 2- and 3-strong pseudoprimes. Users will also have access to an online guide that gives illustrative examples of each function. ... Read more

Customer Reviews (1)

Typesetting ruins a great book
I love the hardcover version of this book and was really excited to see I could get it on the kindle.It was the first thing I ordered and was the first big disappointment.

The typesetting for the math equations makes many of them essentially unreadable with characters overstriking other characters, super and subscripts far from the characters they are super or subscripting, etc..

It appears to me that Amazon is probably using OCR technology that isn't handling this stuff well.Personally I think they should reimburse me for this poorly transcribed version of what is such a great book in the hardcover version although I'm not going to push it. ... Read more