Authors:
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
One of the few books to include the AKS algorithm that shows that primality testing is one of polynomial time
Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers
Includes supplementary material: sn.pub/extras
Buy it now
Buying options
Tax calculation will be finalised at checkout
Other ways to access
This is a preview of subscription content, log in via an institution to check for access.
Table of contents (7 chapters)
-
Front Matter
-
Back Matter
About this book
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.
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)
Authors and Affiliations
-
Fairfield University Dept. Mathematics, Fairfield, USA
Benjamin Fine
-
Universität Hamburg , Dortmund, Germany
Gerhard Rosenberger
About the authors
Benjamin Fine, PhD, is Professor of Mathematics at Fairfield University, CT, USA.
Gerhard Rosenberger, PhD, is Professor (retired) at Dortmund University of Technology, Germany.
Bibliographic Information
Book Title: Number Theory
Book Subtitle: An Introduction via the Density of Primes
Authors: Benjamin Fine, Gerhard Rosenberger
DOI: https://doi.org/10.1007/978-3-319-43875-7
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG 2016
Hardcover ISBN: 978-3-319-43873-3Published: 27 September 2016
Softcover ISBN: 978-3-319-82931-9Published: 14 June 2018
eBook ISBN: 978-3-319-43875-7Published: 19 September 2016
Edition Number: 2
Number of Pages: XIII, 413
Number of Illustrations: 11 b/w illustrations, 1 illustrations in colour
Topics: Number Theory, Mathematical Logic and Foundations, Linear and Multilinear Algebras, Matrix Theory, Analysis, Applications of Mathematics, Data Structures and Information Theory