Introduction to Number Theory

Description

Introduction to Number Theory covers the essential content of an introductory number theory course including divisibility and prime factorization, congruences, and quadratic reciprocity. The instructor may also choose from a collection of additional topics. Aligning with the trend toward smaller, essential texts in mathematics, the author strives for clarity of exposition. Proof techniques and proofs are presented slowly and clearly. The book employs a versatile approach to the use of algebraic ideas. Instructors who wish to put this material into a broader context may do so, though the author introduces these concepts in a non-essential way. A final chapter discusses algebraic systems (like the Gaussian integers) presuming no previous exposure to abstract algebra. Studying general systems helps students to realize unique factorization into primes is a more subtle idea than may at first appear; students will find this chapter interesting, fun and quite accessible. Applications of number theory include several sections on cryptography and other applications to further interest instructors and students alike. Cover Half Title Series Page Title Page Copyright Page Dedication Table of Contents Preface Author Introduction: What is Number Theory? 0.1 Exercises 1 Divisibility 1.1 The Principles of Well-Ordering and Mathematical Induction Exercises 1.2 Basic Properties of Divisibility Exercises 1.3 The Greatest Common Divisor Exercises 1.4 The Euclidean Algorithm Exercises 1.5 Primes Exercises 1.6 Numbers to Different Bases Exercises Challenge Problems for Chapter 1 2 Congruences and Modular Arithmetic 2.1 Basic Definitions and Principles Exercises 2.2 Arithmetic in Z[sub(n)] Exercises 2.3 Linear Equations in Z[sub(n)] Exercises 2.4 The Euler Phi Function Exercises 2.5 Theorems of Wilson, Fermat and Euler Exercises 2.6 Pythagorean Triples Exercises Challenge Problems for Chapter 2 3 Cryptography: An Introduction 3.1 Basic Definitions 3.2 Classical Cryptography Exercises 3.3 Public Key Cryptography: RSA Exercises Challenge Problems for Chapter 3 4 Perfect Numbers 4.1 Basic Definitions and Principles: The Sigma Function Exercises 4.2 Even Perfect Numbers Exercises Challenge Problems for Chapter 4 5 Primitive Roots 5.1 Order of an Integer Exercises 5.2 Primitive Roots Exercises 5.3 Polynomials in Z[sub(p)] Exercises 5.4 Primitive Roots Modulo a Prime Exercises 5.5 An Application: Diffie-Hellman Key Exchange 5.6 Another Application: ElGamal Cryptosystem Challenge Problems for Chapter 5 6 Quadratic Reciprocity 6.1 Squares Modulo a Prime Exercises 6.2 Euler's Criterion and Legendre Symbols Exercises 6.3 The Law of Quadratic Reciprocity Exercises 6.4 The Supplemental Relations Exercises 6.5 The Jacobi Symbol Exercises Challenge Problems for Chapter 6 7 Arithmetic Beyond the Integers 7.1 Gaussian Integers: Introduction and Basic Facts Exercises 7.2 A Geometric Interlude Exercises 7.3 Divisibility and Primes in the Gaussian Integers Exercises 7.4 The Division Algorithm and the Greatest Common Divisor in Z[i] Exercises 7.5 An Application: Sums of Two Squares Exercises 7.6 Another Application: Diophantine Equations Exercises 7.7 A Third Application: Pythagorean Triples Exercises 7.8 Irreducible Gaussian Integers Exercises 7.9 Other Quadratic Extensions Exercises 7.10 Algebraic Numbers and Integers Exercises 7.11 The Quaternions Exercises 7.12 Sums of Four Squares Challenge Problems for Chapter 7 Appendix A: A Proof Primer Appendix B: Axioms for the Integers Appendix C: Basic Algebraic Terminology Bibliography Index