Number Theory

An Introduction via the Density of Primes

Benjamin Fine, Gerhard Rosenberger (Autoren)

Buch | Hardcover

XIII, 413 Seiten

2016 | 2nd ed. 2016
Springer International Publishing (Verlag)
978-3-319-43873-3 (ISBN)

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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.

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

Introduction and Historical Remarks.- Basic Number Theory.- The Infinitude of Primes.- The Density of Primes.- Primality Testing: An Overview.- Primes and Algebraic Number Theory.- The Fields Q_ p of p -adic Numbers: Hensel's Lemma.- References.- Index.

"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)

Erscheinungsdatum	11.10.2016
Zusatzinfo	XIII, 413 p. 12 illus., 1 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	795 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	AKS algorithm • Analysis • Applications of Mathematics • cryptography • Data Structures, Cryptology and Information Theory • Dirichlet's theorem • Elliptic Curve Cryptography • Hensel's lemma • Linear and Multilinear Algebras, Matrix Theory • Mathematical Logic and Foundations • mathematics and statistics • matrix theory • Number Theory • p-adic numbers • Primality Testing • prime number theorem
ISBN-10	3-319-43873-5 / 3319438735
ISBN-13	978-3-319-43873-3 / 9783319438733
Zustand	Neuware