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"It is a unique book that will be influential."
Irving Kaplansky (Mathematical Sciences Research Institute at Berkeley):
"This book has been meticulously prepared. The ample details and motivation will be appreciated by lots of students; the solid punches at the end of each chapter will be appreciated by everybody. It will certainly be a valuable reference for many people, including me. It deserves success with many adoptions as a text."
Kenneth Williams (Carleton University at Ottawa, Canada):
"An extremely well-written and clear presentation of algebraic number theory suitable for beginning graduate students. The many exercises, applications and references are a very valuable feature of the book."
Nikos Tzanakis (University of Iraklion, Crete, Greece): "It's an impressively rich book, from every point of view, written in a very lively style! Besides the standard topics, it also covers admirably a number of others, not easily found in books of this level. The large number of exercises, half of which are solved in special appendices, are extremely valuable for both the teacher and the student. Equally valuable are the numerous appendices on various topics, which make the book thoroughly self-contained. The modern applications of Algebraic Number Theory that are included, will strongly attract students with an "applications-oriented " mind. The many historical remarks make the reading of the book even more pleasant and the numerous tables even more useful. This book deserves a high position in the literature!"
The table of contents on pages vii--viii is essentially self-descriptive of each chapter's contents, so there is no reason for repetition here. Essentially the theme is to move from the level of the algebraic integer in Chapter One, to the arithmetic of algebraic number fields in Chapter Two, then to ideal theory in Chapter Three, which is exploited in Chapter Four in terms of the theory of ramification, augmented by Galois theory, and finally to a closing Chapter Five on reciprocity laws.
We have selectively avoided the presentation of any local theory. For the reader interested in this approach, there are numerous texts from which to choose. One goal of this text is to present a global approach to algebraic number theory, which makes it easier to present this course at an undergraduate level. This approach includes a proof of Dirichlet's Unit Theorem using the geometry of numbers in Chapter Two; Kummer's proof of FLT for regular primes; a proof of the infinitude of irregular primes, using our development of Bernoulli numbers and Bernoulli polynomials in Chapter Three; a proof of the celebrated Kronecker-Weber Theorem (without class field theory), in Chapter Four; and proofs of the cubic, biquadratic and Eisenstein reciprocity laws, with a proof of the Stickelberger Relation to achieve the latter, including such applications as Furtw\"angler's Theorem and Wieferich's Theorem in Chapter Five.
What sets this text apart are the numerous applications to cryptography, which are placed in the last section of each chapter. Thus, not only are all the basics (and more) of algebraic theory of numbers covered, but also applications of the theory to factoring, primality-testing, and public-key cryptosystems. It is only within the preceding three decades that these applications have emerged from an area of mathematics thought to be quite abstract and, as one often hears in reference to algebraic number theory, devoid of any ``real-world" applications. Another goal of this text is to dispell that myth, and hopefully bring the reader to an appreciation, not only of the radiant beauty of algebraic number theory as a part of pure mathematics, but also of the modern-day ``real-world" applications that are so important in our information-based society.
This book is being published at the closing of a century, which has seen a revolution in information-processing and telecommunications. To ever increasing depths, our everyday lives are impacted by interactions that require our sending of digital missives through cyberspace. From the electronic transfer of digital dollars, and the sending of military secrets, to the sending of personal ``e-mail" messages, we are all affected in some way to varying degrees. What is common to all of the above message-sending is the need for keeping the information secret, for ensuring that nobody tampers with that information, and for determining who sent the data. This is where cryptography, the design and implementation of secrecy systems, walks through the door with an extended hand to help us solve these problems. The importance of cryptography to our information-based society will only deepen in the next century. It is time for us to learn as much as we can about it.
In this text, we isolate each cryptographic application in a separate section at the end of each chapter. In this way, the instructor may choose all, some or none of the applications in the course. The applications sections are meant to supplement a course in algebraic number theory by inclusion of various applications to cryptography, especially factoring and primality testing algorithms.
We begin the applications, at the end of Chapter One, with a method, introduced by John Pollard in 1988, for factoring natural numbers using certain cubic integers, the algebraic integers in a cubic field. This method was the genesis of the number field sieve, which we describe at the end of Chapter Two. The first practical implementation of this sieve was published in 1993, and has since superseded all other algorithms for factoring the special type of numbers to which it is suited, described in Section Six of Chapter Two. The means by which cryptography uses public-key encryption to send secure messages, and why factoring and primality testing are so important to it, are illustrated in Section Seven of Chapter Three, where we discuss the Buchmann-Williams public-key encryption scheme, which uses algebraic number theory in complex quadratic fields. In Section Five of Chapter Four, we illustrate primality testing techniques by describing Hendrik Lenstra's method using Artin symbols. These symbols are a motivator for the topic of Chapter Five, where we discuss one of the pinnacles of algebraic number theory, reciprocity laws, in detail. In the final section of the main text, Section Five of Chapter Five, we develop the basics of the theory of elliptic curves and describe Hendrik Lenstra's elliptic curve factoring method, and the elliptic curve primality test that ensues from it. This is a fitting conclusion to the main text since, as we delineate in that section, elliptic curves provided the venue for the solution of FLT, which in turn was the reason for the genesis of algebraic number theory itself.
Given the above, this text may be used as a pure mathematics course in algebraic number theory that can be turned into a more applied presentation by the inclusion of selected topics from the last sections of the chapters. Moreover, the text is made self-contained by the material in the appendices. More on this and other aspects of the text are described in what follows.
The text is accessible to anyone from the senior undergraduate to the research scientist. The main prerequisites are the basics of a first course in abstract algebra, the fundamentals of an introductory course in elementary number theory, and some knowledge of basic matrix theory. In any case, the appendices, as described below, contain a review of all of the requisite background material. Essentially, the mature student, with a knowledge of algebra, can work through the book without any serious impediment.
There are more than eighty biographies of the mathematicians who helped develop algebraic number theory from its inception. These are given in the footnotes woven throughout the text, to give a human face to the mathematics being presented. Our appreciation of mathematics is deepened by a knowledge of the lives of these individuals. The footnote presentation of their lives allows the reader to have immediate information at will, or to treat them as digressions, and access them later without significantly interfering with the main mathematical text at hand. This author has avoided the current convention of gathering notes at the end of each chapter, since the immediacy of information is more important. Furthermore, there is some historical evidence that notes gathered at the end of chapters are either ignored, or are difficult for the reader to correlate with the information given in that chapter, since most readers are reluctant to continuously flip pages to be reminded of the facts being referenced. The footnotes contain not only the bibliographical information cited above, but also other historical data of interest, as well as other information which the discerning reader may want to explore at leisure.
There are ``real-world" applications via cryptography, factoring and primality testing as described above. These are set apart as optional sections at the end of every chapter.
The appendices are given, for the convenience of the reader, to make the text self-contained. Appendix A is a meant as a convenient finger-tip reference for abstract algebra. This review contains the following topics: groups, rings, fields, modules, vector spaces, mappings, polynomial rings, basic matrix theory, Kronecker products, Stirling's formula, Zorn's lemma, Dirichlet's box principle, the multinomial theorem, the Lagrange interpolation formula; and some facts from elementary number theory on the Legendre symbol, the Jacobi symbol, the Kronecker symbol, and a concluding discussion of the Prime Number Theorem.
Appendix B is an overview of sequences and series, including: properties of infinite series, nine convergence tests for infinite series, powers series, Dirichlet series, gamma functions, the functional equation of the Riemann zeta function, and Euler products.
Appendix C is a self-contained overview, with proofs and exercises, of the basics of Galois theory, especially of finite fields and number fields. We begin with the classical approach of Artin and proceed with the development of normal and separable extension fields, splitting fields and ultimately Galois extensions, including an illustrated description of the Fundamental Theorem of Galois Theory, the proof of which concludes Appendix C.
Appendix D consists of the Greek alphabet with English transliteration. Students and research mathematicians alike have need of the latter in symbolic presentations of mathematical ideas. Thus, it is valuable to have a table of the symbols, and their English equivalents readily at hand.
Appendix E has a table of numerous Latin phrases and their English equivalents. Much mathematical discourse uses Latin (including this preface above). Therefore, it will prove useful to have this table available.
For ease of search, we have consecutive numbering, namely object N.m is the m^th object in Chapter N (or Appendix N), exclusive of footnotes and exercises, which are numbered separately and consecutively unto themselves. Thus, for instance, Theorem 1.111 is the 111^th numbered object in Chapter One, exclusive of footnotes and exercises, Exercise 3.116 is the 116^th exercise in Chapter Three, and Footnote 4.9 is the ninth footnote in Chapter Four.
The list of symbols is designed so that the reader may determine, at a glance, on which page the first defining occurrence of a desired notation exists.
The index has nearly one thousand eight-hundred entries, and has been
devised in such a way to ensure that there is maximum ease in getting information
from the text. There is maximum cross-referencing to ensure that the reader
will find ease-of-use, in extracting information, to be paramount.
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Last updated: June 20, 2009