Synopses & Reviews
An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory. A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography. Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to be introduced to some of the main themes in number theory.
Review
From the reviews: "This number theory text is somewhat different than traditional number theory texts. The authors' guiding principle is unique factorization and its consequences. ... This is not a traditional number theory text, but one that tries to guide the reader through the beginnings of the subject towards the modern frontiers. This is helped along by a good sized bibliography plus many problems ... . it might provide an interesting experience when used at the graduate level." (Don Redmond, Mathematical Reviews, Issue 2006 j) "The book under review contains several topics which are usually not brought together in an introductory text. The book is meant to give a broad introduction to advanced undergraduate students ... of number theory. ... Each chapter contains many exercises and historical notes. ... In my opinion, because so many topics are treated in an accessible way, the book is very well suited for an introductory course in number theory." (Jan-Hendrik Evertse, Zentralblatt MATH, Vol. 1089 (15), 2006) "In An Introduction to Number Theory, the authors strive to have the best of all worlds: they cover a broad range of topics ... . This book could be used for a number of different courses. ... The full book would be appropriate for a first-year graduate course. It's also a nice introduction to the subject for established mathematicians form other fields. ... its extensive bibliography, tasteful collection of topics, and clear presentation make it a pleasant reference even for working number theorists." (Rob Benedetto, MathDL, January 2006)
Synopsis
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.
One of the main strengths of this book is the narrative. Everest and Ward present number theory as a living subject, showing how various new developments have drawn upon older traditions.
The authors concentrate on the underlying ideas instead of working out the most general and complete version of a result. They select material from both the algebraic and analytic disciplines and sometimes present several different proofs of a single result to show the differing viewpoints and also to capture the imagination of the reader and help them to discover their own tastes. They also cover important topics of significant interest, eg. elliptic functions and the new primality test, which are often omitted from other books at this level.
Synopsis
This text covers material from both the algebraic and analytic disciplines and includes coverage of recent developments, such as the new primality test and other topics of significant interest, which are often omitted from other books at this level. It aims to take readers to a deeper understanding of the patterns of thought that have shaped the modern understanding of number theory.
Synopsis
Includes up-to-date material on recent developments and topics of significant interest, such as elliptic functions and the new primality test Selects material from both the algebraic and analytic disciplines, presenting several different proofs of a single result to illustrate the differing viewpoints and give good insight
Table of Contents
A Brief History of Prime.- Diophantine Equations.- Quadratic Diophantine Equations.- Recovering the Fundamental Theorem of Arithmetic.- Elliptic Curves.- Elliptic Functions.- Heights.- The Riemann Zeta Function.- The Functional Equation of the Riemann Zeta Function.- Primes in an Arithmetic Progression.- Converging Streams.- Computational Number Theory.- References.- Index.