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Hardcover ISBN:  9781470450168 
Product Code:  AMSTEXT/35 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470451905 
Product Code:  AMSTEXT/35.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470450168 
eBook ISBN:  9781470451905 
Product Code:  AMSTEXT/35.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 35; 2019; 488 ppMSC: Primary 11; 14
Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. 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 find 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 first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the juniorsenior level.
ReadershipUndergraduate and graduate students interested in learning and teaching.

Table of Contents

Cover

Title page

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 𝑝

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)^{×}

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 𝑝Adic 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 𝑥²𝐵𝑦²=1

14.3. Generalized Pell’s Equations 𝑥²𝐵𝑦²=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

Back Cover


Additional Material

Reviews

I think this book would be a great foundation for a course which is more inspiring—and perhaps more challenging—than your standard course on elementary number theory.
Abbey Bourdon, MAA Reviews


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Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. 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 find 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 first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the juniorsenior level.
Undergraduate and graduate students interested in learning and teaching.

Cover

Title page

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 𝑝

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)^{×}

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 𝑝Adic 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 𝑥²𝐵𝑦²=1

14.3. Generalized Pell’s Equations 𝑥²𝐵𝑦²=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

Back Cover

I think this book would be a great foundation for a course which is more inspiring—and perhaps more challenging—than your standard course on elementary number theory.
Abbey Bourdon, MAA Reviews