Number Theory

1. Front Matter

   Pages i-xiii

   PDF

2. Introduction and Historical Remarks

   • Benjamin Fine, Gerhard Rosenberger

   Pages 1-5

3. Basic Number Theory

   • Benjamin Fine, Gerhard Rosenberger

   Pages 7-58

4. The Infinitude of Primes

   • Benjamin Fine, Gerhard Rosenberger

   Pages 59-142

5. The Density of Primes

   • Benjamin Fine, Gerhard Rosenberger

   Pages 143-218

6. Primality Testing—An Overview

   • Benjamin Fine, Gerhard Rosenberger

   Pages 219-284

7. Primes and Algebraic Number Theory

   • Benjamin Fine, Gerhard Rosenberger

   Pages 285-370

8. The Fields \({\mathbb {Q}}_p\) of p-Adic Numbers: Hensel’s Lemma

   • Benjamin Fine, Gerhard Rosenberger

   Pages 371-404

9. Back Matter

   Pages 405-413

   PDF

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.

• AKS algorithm
• Cryptography
• Dirichlet's Theorem
• Hensel's lemma
• Number Theory
• Primality Testing
• Prime Number Theorem
• p-adic Numbers
• Elliptic Curve Cryptography
• matrix theory

“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)

• Fairfield University Dept. Mathematics, Fairfield, USA

  Benjamin Fine

• Universität Hamburg , Dortmund, Germany

  Gerhard Rosenberger

Benjamin Fine, PhD, is Professor of Mathematics at Fairfield University, CT, USA.

Gerhard Rosenberger, PhD, is Professor (retired) at Dortmund University of Technology, Germany.

Policies and ethics