Overview

Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing. Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.

Full Product Details

Author:   Benjamin Fine ,  Gerhard Rosenberger
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   Softcover reprint of the original 2nd ed. 2016
Dimensions:   Width: 15.50cm , Height: 2.20cm , Length: 23.50cm
Weight:   0.652kg
ISBN:  

9783319829319

ISBN 10:   3319829319
Pages:   413
Publication Date:   14 June 2018
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Reviews

In this text, Fine (mathematics, Fairfield Univ.) and Rosenberger (Univ. of Hamburg, Germany) successfully present number theory from the inception of primes to recent developments in algebraic and analytic number theory and cryptography. ... Numerous exercises and open problems are provided. The breadth and depth of topics covered are impressive, making this an excellent text for those interested in the field of number theory. Summing Up: Recommended. Upper-division undergraduates and graduate students. (J. T. Zerger, Choice, Vol. 54 (9), May, 2017) The book is chatty and leisurely, with lots of historical notes and lots of worked examples. The exercises at the end of each chapter are good and there are a reasonable number of them. ... a good text for an introductory course ... . (Allen Stenger, MAA Reviews, maa.org, November, 2016)

Author Information

Benjamin Fine, PhD, is Professor of Mathematics at Fairfield University, CT, USA. Gerhard Rosenberger, PhD, is Professor (retired) at Dortmund University of Technology, Germany.

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