Explorations in Number. Theory Commuting through the Numberverse

Description

Preface Who is This Text’s Audience? To the Student: To the Instructor: Suggested Pacing and Content Coverage Contents 1 What is a Number? 1.1 Human conception of numbers 1.2 Algebraic Number Systems 1.3 New Numbers, New Worlds 1.4 Exercises 2 A Quick Survey of the Last Two Millennia 2.1 Fermat, Wiles, and The Father of Algebra 2.2 Quadratic Equations 2.3 Diophantine Equations 2.4 Exercises 3 Number Theory in Z Beginning 3.1 Algebraic Structures 3.2 Linear Diophantine Equations and the Euclidean Algorithm 3.3 The Fundamental Theorem of Arithmetic 3.4 Factors and Factorials 3.5 The Prime Archipelago 3.6 Exercises 4 Number Theory in the Mod-n Era 4.1 Equivalence Relations and the Binary World 4.2 The Ring of Integers Modulo n 4.3 Reduce First and ask Questions Later 4.4 Division, Exponentiation, and Factorials in Zn 4.5 Group Theory and the Ring of Integers Modulo n 4.6 Lagrange's Theorem and the Euler Totient Function 4.7 Sunzi's Remainder Theorem and phi(n) 4.8 Phis, Polynomials, and Primitive Roots 4.9 Exercises 5 Gaussian Number Theory: Zi of the Storm 5.1 The Calm Before 5.2 Gaussian Divisibility 5.3 Gaussian Modular Arithmetic 5.4 Gaussian Division Algorithm: The Geometry of Numbers 5.5 A Gausso-Euclidean Algorithm 5.6 Gaussian Primes and Prime Factorizations 5.7 Applications to Diophantine Equations 5.8 Exercises 6 Number Theory, from Where We R to Across the C 6.1 From -1 to -d 6.2 Algebraic Numbers and Rings of Integers 6.3 Quadratic Fields: Integers, Norms, and Units 6.4 Euclidean Domains 6.5 Unique Factorization Domains 6.6 Euclidean Rings of Integers 6.7 Exercises 7 Cyclotomic Number Theory: Roots and Reciprocity 7.1 Introduction 7.2 Quadratic Residues and Legendre Symbols 7.3 Quadratic Residues and Non-Residues Mod p 7.4 Application: Counting Points on Curves 7.5 The Quadratic Reciprocity Law: Statement and Use 7.6 Some Unexpected Helpers: Roots of Unity 7.7 A Proof of Quadratic Reciprocity 7.8 Quadratic UFDs 7.9 Exercises 8 Number Theory Unleashed: Release Zp 8.1 The Analogy between Numbers and Polynomials 8.2 The p-adic World: An Analogy Extended 8.3 p-adic Arithmetic: Making a Ring 8.4 Which numbers are p-adic? 8.5 Hensel's Lemma 8.6 The Local-Global Philosophy and the Infinite Prime 8.7 The Local-Global Principle for Quadratic Equations 8.8 Computations: Quadratic Equations Made Easy 8.9 Synthesis and Beyond: Moving Between Worlds 8.10 Exercises 9 The Adventure Continues 9.1 Exploration: Fermat's Last Theorem for Small n 9.2 Exploration: Lagrange's Four-Square Theorem 9.3 Exploration: Public Key Cryptography 9.3.1 Public Key Encryption: RSA 9.3.2 Elliptic Curve Cryptography 9.3.3 Elliptic ElGamal Public Key Cryptosystem 9.4 Exploration: Units of Real Quadratic Fields 9.5 Exploration: Ideals and Ideal Numbers 9.6 Conclusion: The Numberverse, Redux Appendix I Number Systems I.1 Introduction I.2 Construction of the Natural Numbers I.3 Induction and Well-Ordering Appendix Index Index Appendix Index of Notation Author Index