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/m\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 pⁿ Elements
6.5. Fields with p² Elements
6.6. Fields with s 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 𝑝^{𝑘}
8.6. Indices
8.7. Existence of Primitive Roots Modulo m
8.8. The Structure of (\Z/𝑝^{𝑘}\Z)^{x}
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 p-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 n=4
12.4. Ellipses
12.5. Quadratic Fields and Norms
12.6. Integral Points on Ellipses
12.7. Primes of the Form X²+BY²
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 x²-By²=1
14.3. Generalized Pell's Equations x²-By²=N
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 E(\Q)
16.4. The Torsion Subgroup
16.5. Elliptic Curves over Finite Fields
16.6. The Rank and the Free Part of E(\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