This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to ?nd all the integral points on a line in the plane. Similarly, Gauss’s law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively.
This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a ?rst course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior- senior level.
Álvaro Lozano-Robledo: University of Connecticut, Storrs, CT
Preface
Chapter 1. Introduction 1.1. Roots of Polynomials 1.2. Lines 1.3. Quadratic Equations and Conic Sections 1.4. Cubic Equations and Elliptic Curves 1.5. Curves of Higher Degree 1.6. Diophantine Equations 1.7. Hilbert’s Tenth Problem 1.8. Exercises Part 1 . Integers, Polynomials, Lines, and Congruences
Chapter 2. The Integers 2.1. The Axioms of \Z 2.2. Consequences of the Axioms 2.3. The Principle of Mathematical Induction 2.4. The Division Theorem 2.5. The Greatest Common Divisor 2.6. Euclid’s Algorithm to Calculate a GCD 2.7. Bezout’s Identity 2.8. Integral and Rational Roots of Polynomials 2.9. Integral and Rational Points in a Line 2.10. The Fundamental Theorem of Arithmetic 2.11. Exercises
Chapter 3. The Prime Numbers 3.1. The Sieve of Eratosthenes 3.2. The Infinitude of the Primes 3.3. Theorems on the Distribution of Primes 3.4. Famous Conjectures about Prime Numbers 3.5. Exercises
Chapter 4. Congruences 4.1. The Definition of Congruence 4.2. Basic Properties of Congruences 4.3. Cancellation Properties of Congruences 4.4. Linear Congruences 4.5. Systems of Linear Congruences 4.6. Applications 4.7. Exercises
Chapter 5. Groups, Rings, and Fields 5.1. \Z/??\Z 5.2. Groups 5.3. Rings 5.4. Fields 5.5. Rings of Polynomials 5.6. Exercises
Chapter 6. Finite Fields 6.1. An Example 6.2. Polynomial Congruences 6.3. Irreducible Polynomials 6.4. Fields with ??n Elements 6.5. Fields with ??² Elements 6.6. Fields with ?? Elements 6.7. Exercises
Chapter 7. The Theorems of Wilson, Fermat, and Euler 7.1. Wilson’s Theorem 7.2. Fermat’s (Little) Theorem 7.3. Euler’s Theorem 7.4. Euler’s Phi Function 7.5. Applications 7.6. Exercises
Chapter 8. Primitive Roots 8.1. Multiplicative Order 8.2. Primitive Roots 8.3. Universal Exponents 8.4. Existence of Primitive Roots Modulo ?? 8.5. Primitive Roots Modulo ??^{??} 8.6. Indices 8.7. Existence of Primitive Roots Modulo ?? 8.8. The Structure of (\Z/??^{??}\Z)^{×} 8.9. Applications 8.10. Exercises Part 2 . Quadratic Congruences and Quadratic Equations Chapter 9. An Introduction to Quadratic Equations 9.1. Product of Two Lines 9.2. A Classification: Parabolas, Ellipses, and Hyperbolas 9.3. Rational Parametrizations of Conics 9.4. Integral Points on Quadratic Equations 9.5. Exercises
Chapter 10. Quadratic Congruences 10.1. The Quadratic Formula 10.2. Quadratic Residues 10.3. The Legendre Symbol 10.4. The Law of Quadratic Reciprocity 10.5. The Jacobi Symbol 10.6. Cipolla’s Algorithm 10.7. Applications 10.8. Exercises
Chapter 11. The Hasse–Minkowski Theorem 11.1. Quadratic Forms 11.2. The Hasse–Minkowski Theorem 11.3. An Example of Hasse–Minkowski 11.4. Polynomial Congruences for Prime Powers 11.5. The ??-Adic Numbers 11.6. Hensel’s Lemma 11.7. Exercises
Chapter 12. Circles, Ellipses, and the Sum of Two Squares Problem 12.1. Rational and Integral Points on a Circle 12.2. Pythagorean Triples 12.3. Fermat’s Last Theorem for ??=4 12.4. Ellipses 12.5. Quadratic Fields and Norms 12.6. Integral Points on Ellipses 12.7. Primes of the Form ??²+????² 12.8. Exercises
Chapter 13. Continued Fractions 13.1. Finite Continued Fractions 13.2. Infinite Continued Fractions 13.3. Approximations of Irrational Numbers 13.4. Exercises
Chapter 14. Hyperbolas and Pell’s Equation 14.1. Square Hyperbolas 14.2. Pell’s Equation ??²-????²=1 14.3. Generalized Pell’s Equations ??²-????²=?? 14.4. Exercises Part 3 . Cubic Equations and Elliptic Curves
Chapter 15. An Introduction to Cubic Equations 15.1. The Projective Line and Projective Space 15.2. Singular Cubic Curves 15.3. Weierstrass Equations 15.4. Exercises
Chapter 16. Elliptic Curves 16.1. Definition 16.2. Integral Points 16.3. The Group Structure on ??(\Q) 16.4. The Torsion Subgroup 16.5. Elliptic Curves over Finite Fields 16.6. The Rank and the Free Part of ??(\Q) 16.7. Descent and the Weak Mordell–Weil Theorem 16.8. Homogeneous Spaces 16.9. Application: The Elliptic Curve Diffie–Hellman Key Exchange 16.10. Exercises
Bibliography Index