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This classic book gives a thorough introduction to constructive algebraic number theory, and is therefore especially suited as a textbook for a course on that subject. It also provides a comprehensive look at recent research. For experimental number theoreticians, the authors developed new methods and obtained new results of great importance for them. Both computer scientists interested in higher arithmetic and those teaching algebraic number theory will find the book of value. ... Read more

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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

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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 andmany explicit code constructions based on algebraic number theory 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

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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. Contents: Introduction • Topics from finite fields • Arithmetic and polynomials • Factorization of polynomials • Topics from the geometry of numbers • Hermite normal form • Lattices • Reduction • Enumeration of lattice points • Algebraic number fields • Introduction • Basic Arithmetic • Computation of an integral basis • Integral closure • Round-Two-Method • Round-Four-Method • Computation of the unit group • Dirichlet's unit theorem and a regulator bound • Two methods for computing r independent units • Fundamental unit computation • Computation of the class group • Ideals and class number • A method for computing the class group • Appendix • The number field sieve • KANT • References • Index ... Read more

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These proceedings contain 35 selected research and survey articles based on lectures delivered at the International Conference on Algebraic Number Theory and Diophantine Analysis held in Graz, Austria, August 30 – September 5, 1998. Although the conference was open for all branches of number theory, there was a special emphasis on algebraic number theory, on diophantine and algorithmic problems and on diophantine analysis, uniform distribution and discrepancy theory. Several lectures were devoted to the effective and algorithmic solution of diophantine equations and related topics.

This volume is of interest to researchers and graduate students in number theory including applications, in computer algebra and dynamical systems. ... Read more

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This volume contains the proceedings of a seminar on Algebraic $K$-theory and Algebraic Number Theory, held at the East-West Center in Honolulu in January 1987. The seminar, which hosted nearly 40 experts from the U.S. and Japan, was motivated by the wide range of connections between the two topics, as exemplified in the work of Merkurjev, Suslin, Beilinson, Bloch, Ramakrishnan, Kato, Saito, Lichtenbaum, Thomason, and Ihara. As is evident from the diversity of topics represented in these proceedings, the seminar provided an opportunity for mathematicians from both areas to initiate further interactions between these two areas. ... Read more

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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. ... Read more

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A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher. ... Read more

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This scarce antiquarian book is included in our special Legacy Reprint Series. In the interest of creating a more extensive selection of rare historical book reprints, we have chosen to reproduce this title even though it may possibly have occasional imperfections such as missing and blurred pages, missing text, poor pictures, markings, dark backgrounds and other reproduction issues beyond our control. Because this work is culturally important, we have made it available as a part of our commitment to protecting, preserving and promoting the world's literature. ... Read more

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This book presents papers that originally appeared in the Japanese journal Sugaku from the Mathematical Society of Japan. The papers explore the relationship between number theory and algebraic geometry. ... Read more

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This book presents papers that originally appeared in the Japanese journal Sugaku. The papers explore the relationship between number theory, algebraic geometry, and differential geometry. ... Read more