Book Description
This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten. ... Read more

Book Description
...reflects the exciting developments in number theory during the past two decades. ... Read more

Customer Reviews (4)

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.

thoughts from an amateur
good overview of algebraic number theory as it applies to FLT, however not exactly pitched at beginners.you'll want to have a grounding in abstract algebra & linear algebra at the minimum.still, even if you don't, you can get a good sense of the "big picture" and a high-level understanding of the advances in mathematics that were directly or indirectly related to attempts to solve FLT.overall a fascinating read if you're a math geek who wants something a little deeper than Simon Singh's pop treatment of Wiles' proof.

Lucid introduction
Lucid and clear introduction to algebraic number theory, in style very much like the author's other book on Galois theory. Very elementary though, doesn't cover any analytic method, nor gives even a taste of class field theory, besides the problem set is less than challenging. But the book serves its purpose well, strongly recommended for beginners. ... Read more

Book Description
Asking how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. However, it is a recognized fact that problem solving plays an important role in training the mind of a researcher. It would not be an exaggeration to say that the ability to do mathematical research lies essentially asking "well-posed" questions. The approach taken by the authors in Problems in Algebraic Number Theory is based on the principle that questions focus and orient the mind. The book is a collection of about 500 problems in algebraic number theory, systematically arranged to reveal ideas and concepts in the evolution of the subject. While some problems are easy and straightforward, others are more difficult. For this new edition the authors added a chapter and revised several sections. The text is suitable for a first course in algebraic number theory with minimal supervision by the instructor. The exposition facilitates independent study, and students having taken a basic course in calculus, linear algebra, and abstract algebra will find these problems interesting and challenging. For the same reasons, it is ideal for non-specialists in acquiring a quick introduction to the subject. ... Read more

Customer Reviews (3)

Excellent book on problems
This is a very useful book for anyone studying number theory. It's especially helpful for amatuer mathematicians learning on their own. This one is the same as the older edition with more hints and more detailed explanation. But WOULDN'T IT BE GREAT TO LEAVE A little room for the readers to think on their own?! You will reap the benefit from thinking hard as well as working hard!

Amazing!!
Best book of it's kind that I've ever read.I found it to be extremely helpful.I even read it at night before I go to bed because its so entertaining.I thoroughly enjoyed it and would recommend it to anyone who is studying this.

Won't become a classic.
A problem book is always helpful to students. But this one is sloppy. Besides the supplementary problems at the end of each chapter, most problems are boring or break-ups of theorems, and there isn't much enlightment or warmth or lucidity in presentation of the materials. I do find though, the author's other book, "Problems in analytic number theory", is far superior to this one. ... Read more

Book Description
Computational algebraic number theory has been attracting broad interest in the last few years due to its potential applications in coding theory and cryptography. For this reason, the Deutsche Mathematiker Vereinigung initiated an introductory graduate seminar on this topic in Düsseldorf. The lectures given there by the author served as the basis for this book which allows fast access to the state of the art in this area. Special emphasis has been placed on practical algorithms - all developed in the last five years - for the computation of integral bases, the unit group and the class group of arbitrary algebraic number fields. ... Read more

Book Description

Book Description

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

Book Description
Algebraic number theory is gaining an increasing impact in code design for many different coding applications, such assingle antenna fading channels and more recently, MIMO systems.Extended work has been done on single antenna fading channels,and algebraic lattice codes have been proven to be an effective tool. The general framework has been developed in the last ten years and many explicit code constructions based on algebraic numbertheory are now available.Algebraic Number Theory and Code Design for Rayleigh Fading Channels provides an overview ofalgebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory.The basic facts of this mathematical field are illustrated by many examples and by the use of computer algebra freeware in order to make it more accessible to a large audience. This makes the book suitable for use by students and researchers in both mathematics and communications. ... Read more

Book Description
This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations. ... Read more

Customer Reviews (1)

Technical, but concrete
It is an unfortunate feature of number theory that few of the books explain clearly the motivation for much of the technology introduced.Similarly, half of this book is spent proving properties of Dedekind domains before we see much motivation.

That said, there are quite a few examples, as well as some concrete and enlightening exercises (in the back of the book, separated by chapter).There is also a chapter, if the reader is patient enough for it, on Diophantine equations, which gives a good sense of what all this is good for.

The perspective of the book is global.Central themes are the calculation of the class number and unit group.The finiteness of the class number and Dirichlet's Unit Theorem are both proved.L-functions are also introduced in the final chapter.

While the instructor should add more motivation earlier, the book is appropriate for a graduate course in number theory, for students who already know, for instance, the classification of finitely generated modules over a PID.It may be better than others, but would be difficult to use for self-study without additional background. ... Read more

Book Description
Contents: Preface. Reader's Guide. Index of Notation. The Origins of Algebraic Number Theory. Part I: Numbers. Quadratic and Cyclotomic Fields. Geometric Methods. Lattices. Minkowski's Theorem. Part II: Geometric Representation of Algebraic Numbers. Class-Group and Class-Number. Part III: Number-Theoretic Applications. Computational Methods. Fermat's Last Theorem. Dirichlet's Units Theorem. Appendix. Quadratic Residues. References. Index. ... Read more

Book Description
From the review: "The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further study, and concrete examples.... The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook.... The book is, without any doubt, the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available."
W. Kleinert in: Zentralblatt für Mathematik, 1992 ... Read more

Customer Reviews (2)

One of the most beautifully written math books
This book is basically all you need to learn modern algebraic number theory. You need to know algebra at a graduate level (Serge Lang's Algebra) and I would recommend first reading an elementary classical Algebraic number theory book like Ian Stewart's Algebraic Number Theory, or Murty and Esmonde's Problem's in Algebraic Number theory.

10 stars if I could.
After having no fun with Lang's text "Algebraic Number Theory" I began seking out something more complete and which was full of quality exposition.As a result of Amazon's approach to marketing towards members, I was recommended this book and decided quickly that I must have it.This book is marvelously well written, examples are kept to an un-overwhelming minimum, the problems are not trivial (at least to me) and in fact I feel this is the kind of book on par with, say, Paulo Ribenboim's "Classical Theory of Algebraic Numbers" since these are both the type of book you would want to take with you on a long trip or as Paulo says, "while stranded on a desert island".This book is by no means intended for those who are not fluent in both Number Theory as well as Algebra, both at the graduate level and obviously for those who are Mahematically gifted.I highly recommend this book to graduate students interested in Algebraic number theory as well as those needing a splendid reference. ... Read more

Book Description
Suitable for senior undergraduates and beginning graduate students in mathematics, this book is an introduction to algebraic number theory at an elementary level. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested readings and to the biographies of mathematicians who have contributed to the development of algebraic number theory are provided at the end of each chapter.Other features include over 320 exercises, an extensive index, and helpful location guides to theorems in the text. ... Read more

Customer Reviews (1)

Grab bag of good and bad
Strengths:
1. Easy reading, detailed proofs
2. Covered required algebra background (modules, ideals, Dedekind domains, etc)
3. Many, many examples

Weaknesses:
1. Too detailed in some cases
2. Does not develop more advanced ideas that actually make the material easier
3. Poor index
4. Examples are often too simple

This book takes the reader through the required algebra background and moves them into the realm of using these abstract algebraic construction to study the theory of numbers. The book is aimed at upper-level undergraduates, so it's easy reading. Sometimes too easy reading, as proofs are often long-winded and contain many trivial details. In some instances, I wanted all those details, often it was simply annoying.

The real strength of this book lies in the many explicit examples. It was worth the price for these examples, as most higher-level books offer few examples.

The index is terrible, but the additional reading section at the end of each chapter is a nice addition.

Overall, I learned a lot from this book, but would have liked to have the authors approached the material at a little bit higher level. For instance, instead of using complex conjugates extensively, I would have preferred introducing a mapping to the complex conjugates (say sigma) for use in most proofs. ... Read more

Book Description
From the reviews: "... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." Monatshefte fuer Mathematik, 1994"... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." Medelingen van het wiskundig genootschap, 1995 ... Read more

Book Description
This book is a translation into English of Hilbert's "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provided an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht provided also a firm foundation for further research in the subject. It is based on the work of the great number theorists of the nineteenth century. The Zahlbericht can be seen as the starting point of all twentieth century investigations in algebraic number theory, reciprocity laws and class field theory. For this English edition an Introduction has been added by F. Lemmermeyer and N. Schappacher. ... Read more

Book Description
This account of Algebraic Number Theory is written primarily for beginning graduate students in pure mathematics, and encompasses everything that most such students are likely to need; others who need the material will also find it accessible. It assumes no prior knowledge of the subject, but a firm basis in the theory of field extensions at an undergraduate level is required, and an appendix covers other prerequisites. The book covers the two basic methods of approaching Algebraic Number Theory, using ideals and valuations, and includes material on the most usual kinds of algebraic number field, the functional equation of the zeta function and a substantial digression on the classical approach to Fermat's Last Theorem, as well as a comprehensive account of class field theory. Many exercises and an annotated reading list are also included. ... Read more

Book Description
This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book. ... Read more

Customer Reviews (3)

Old school algebraic number theory with heavy Kummer bias
Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

great book
This is a great book.If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want.it really has a lot of stuff in it.all other graduate books are theory without examples or motivation.this book is the exact opposite.the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

Read this if you're seriously interested in math.
There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book. ... Read more

Book Description
From its history as an elegant but abstract area of mathematics, algebraic number theory now takes its place as a useful and accessible study with important real-world practicality. Unique among algebraic number theory texts, this important work offers a wealth of applications to cryptography, including factoring, primality-testing, and public-key cryptosystems.A follow-up to Dr. Mollin's popular Fundamental Number Theory with Applications, Algebraic Number Theory provides a global approach to the subject that selectively avoids local theory. Instead, it carefully leads the student through each topic from the level of the algebraic integer, to the arithmetic of number fields, to ideal theory, and closes with reciprocity laws. In each chapter the author includes a section on a cryptographic application of the ideas presented, effectively demonstrating the pragmatic side of theory.In this way Algebraic Number Theory provides a comprehensible yet thorough treatment of the material. Written for upper-level undergraduate and graduate courses in algebraic number theory, this one-of-a-kind text brings the subject matter to life with historical background and real-world practicality. It easily serves as the basis for a range of courses, from bare-bones algebraic number theory, to a course rich with cryptography applications, to a course using the basic theory to prove Fermat's Last Theorem for regular primes. Its offering of over 430 exercises with odd-numbered solutions provided in the back of the book and, even-numbered solutions available a separate manual makes this the ideal text for both students and instructors. ... Read more

Customer Reviews (3)

A Mature and Concrete Introduction to Algebraic Number Theory
I used this book in a one on one course in algebraic number theory in my fourth year of college.We finished just before ramification.

My background at the time included a year of undergraduate linear algebra, a year of undergraduate abstract algebra, a semester of intro. graduate algebra, intro. Galois theory, and intro. commutative algebra. The only things I used were Galois theory, my second abstract algebra course, and my second linear algebra course.Commutative algebra helped, but wasn't necessary in that abstract setting.

Organization: The book is very well organized with helpful appendices on abstract algebra basics (Groups, Rings, Fields) and Galois theory. The first chapter is slow-paced and provides a strong historical background for the material.A reviewer below suggested that there were "logical leaps" in the text--I never found such stuff, and I am always very picky about details. The author uses easy propositions that are assigned for homework sometimes, but they're mostly straightforward.

Exercises: They range from straightforward to quite thought-invoking... I remember one in particular, a starred problem, in which I had to use three "tricks" to solve.

Content: I like this book a lot.It's not super abstract on the level of Lang, but has hints of great generality throughout, and it's not some trivial algebraic number theory full of history, anecdotes, useless junk book with "Fermat's Last Theorem" misleadingly stated in the title somewhere.This book has a lot of stuff on applications to cryptography. The book covers things including

- Euclidean domains and unique factorization,
- special cases of FLT,
- Dirichlet's Unit Theorem,
- geometry of numbers,
- ideal class group,
- ramification,
- basics of class field theory,
- reciprocity laws.

It's a really nice all-around introduction. You need to be mature enough to read this book--the problems require that the reader is familiar with the relevent math. I was really impressed with the organization of this book--with topics like these, it's hard to have a nice balance of generality with concrete and useful results. This is for people not ready to appreciate Lang's book.

Worst textbook ever!
I am a graduate student specializing in Ring Theory and I have to tell you this is the absolute worst book I have ever had. Not only does the author make these humongous jumps in each section, he also has massive logical gaps. There are plenty of errors in the text starting right from the first section. You could easily spend a whole year deciphering (with a massive headache) the first chapter. The author is definetly wrong in assuming that all you need is a basic undergraduate number theory class and basic abstract algebra. You could have 2 comprehensive years of graduate modern algebra and still not be ready for the massive logical gaps in the book. Sure if we were all Galois and Eulers, the book would be easier, but I'd bet they'd even be scratching their heads often enough. My advice is stay away from the text at all cost. You'll regret paying the outrageous price for a text that is worth firepaper.

BUY THIS BOOK!
I learned a tremendous amount about Algebraic Number Theory from this excellent source. I have looked at other books that just skim the topics. This one covers them in depth and even has applications to cryptography (the author shuld have put that in the title). Even more advanced topics such as the higher reciprocity laws are covered with rigorous detail and extreme clarity. I read the AMS review for this book by Charles Parry and it is right on! This book should replace the old standards such as Janusz's and Marcus' books for instance. I'd say that this is a gem to be enjoyed. ... Read more

Book Description
This book describes 148 algorithms which are fundamental for number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters lead the reader to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations. The last three chapters give a survey of factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The book ends with a description of available computer packages and some useful tables. The book also contains a large number of exercises. Written by an authority in the field, and one with great practical and teaching experience it is sure to become the standard and indispensable reference on the subject. ... Read more

Customer Reviews (3)

Great book for computational aspects
I bought this book for the math course I had taken having the same title. This is an excellent book, but only if you are really interested in its content. It's not a casual read, since it's graduate text. Also, a background in number theory will be of great help - being a CS major, I had a little tough time in the beginning, but things turned out just fine. As for content, it has excellent coverage of the subject, and I would highly recommend this as a reference in this subject. Remember, though, that this book deals COMPUTATIONAL aspects, for only number theory, look for other books like Ireland-Rosen.

Definitely belongs on the shelf of all number theory lovers
This book is an excellent compilation of both the theory and pseudo-code for number theoretic algorithms. The author also takes the time to prove some of the major results as background to the algorithms, in addition to sets of exercises at the end of the book. The book is too large to do a chapter by chapter review, so instead I will list the algorithms in the book that I thought were particularly useful:

1. Most of the algorithms on elliptic curves. The author reminds the reader that number-theoretical experiments resulted in the famous Swinnerton-Dyer Conjecture and the Birch Conjecture. (a) the reduction algorithm, which for a given point in the upper half plane, gives the unique point in the half plane equivalent to this point under the action of the special linear group along with the matrix that maps these two points to each other. (b) The computation of the coefficient g2 and g3 of the Weierstrass equation of an elliptic curve. (c) The computation of the Weierstrass function and its derivative. (d) Determination of the periods of an elliptic curve over the real numbers. (e) The determination of the elliptic logarithm. (f) The reduction of a general cubic (f) The Shanks-Mestre algorithm for computing the order of an elliptic curve over a finite field F(p), where p is prime and greater than 457. (g) The reduction of an elliptic curve modulo p for p > 3. (h) The reduction of an elliptic curve modulo 2 or 3. (i) Reduction of an elliptic curve over the rational numbers. (j) Determination of the rational torsion points of an elliptic curve. (k) Computation of the Hilbert class polynomials and thus a determination of the j-function of an elliptic curve.

2. A few of the algorithms on factoring. (a) The Pollard algorithm for finding non-trivial factors of composites. (The author does not give the improved algorithm due to P. Montgomery, but does give references) (b) Shanks Square Form Factorization algorithm for finding a non-trivial factor of an odd integer. (c) Lenstra's Elliptic Curve test for compositeness.

3. Primality tests (a) The Jacobi Sum Primality Test for a positive integer. (b) Goldwasser-Killian elliptic curve test for a positive integer not equal to 1 and coprime to 6.

The author gives an overview of the computer packages used for number theory, including Pari, which was written by him and his collaborators. I have not used this package, but instead use Lydia and Mathematica for most of the number theoretic computations I need to do.

Excellent!
Cohen (the world renowned expert) starts with the most basic of algorithms(i.e. Euclid & Shanks). He moves seamlessly into Linear Algebra &Polynomials (bedrocks of most CAS). Although meant to be concise, heproves, or sketches a proof of the important results. Finally, the meat ofthe book, C.A.N.T. One important problem is finding the "classnumber" (has to do with unique factorization, which we are allaccustomed to in Z). A detailed description of the continued fractionalgorithm (for finding the fundamental unit), and others made it veryenlightening. He then deals with primality testing and factoring, two veryimportant problems, the latter because of RSA. First, a description of thealgorithm, then the theory behind it. He covered everything, from TrialDivision (Dark Ages) to Pollard Rho to NFS (cutting-edge). Also includedare some useful tables.

Of course, CAS information from 1993, won't bethat helpful (look in his newest, Advanced Topics inC.A.N.T.).

Excellent. Also try Knuth's "Semi-numericalAlgorithms" for a more computer oriented approach. ... Read more

Book Description
Modern number theory, according to Hecke, dates from Gauss's quadratic reciprocity law. The various extensions of this law and the generalizations of the domains of study for number theory have led to a rich network of ideas, which has had effects throughout mathematics, in particular in algebra. This volume of the Encyclopaedia presents the main structures and results of algebraic number theory with emphasis on algebraic number fields and class field theory. Koch has written for the non-specialist. He assumes that the reader has a general understanding of modern algebra and elementary number theory. Mostly only the general properties of algebraic number fields and related structures are included. Special results appear only as examples which illustrate general features of the theory. A part of algebraic number theory serves as a basic science for other parts of mathematics, such as arithmetic algebraic geometry and the theory of modular forms. For this reason, the chapters on basic number theory, class field theory and Galois cohomology contain more detail than the others. This book is suitable for graduate students and research mathematicians who wish to become acquainted with the main ideas and methods of algebraic number theory. ... Read more

Book Description

In this, one of the first books to appear in English on the theory of numbers, the eminent mathematician Hermann Weyl explores fundamental concepts in arithmetic. The book begins with the definitions and properties of algebraic fields, which are relied upon throughout. The theory of divisibility is then discussed, from an axiomatic viewpoint, rather than by the use of ideals. There follows an introduction to p-adic numbers and their uses, which are so important in modern number theory, and the book culminates with an extensive examination of algebraic number fields.

Weyl's own modest hope, that the work "will be of some use," has more than been fulfilled, for the book's clarity, succinctness, and importance rank it as a masterpiece of mathematical exposition.