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Number Theory and Geometry: An Introduction to Arithmetic Geometry
 
Álvaro Lozano-Robledo University of Connecticut, Storrs, CT
Number Theory and Geometry
Hardcover ISBN:  978-1-4704-5016-8
Product Code:  AMSTEXT/35
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
Sale Price: $64.35
eBook ISBN:  978-1-4704-5190-5
Product Code:  AMSTEXT/35.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Sale Price: $55.25
Hardcover ISBN:  978-1-4704-5016-8
eBook: ISBN:  978-1-4704-5190-5
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
Sale Price: $119.60 $91.98
Number Theory and Geometry
Click above image for expanded view
Number Theory and Geometry: An Introduction to Arithmetic Geometry
Álvaro Lozano-Robledo University of Connecticut, Storrs, CT
Hardcover ISBN:  978-1-4704-5016-8
Product Code:  AMSTEXT/35
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
Sale Price: $64.35
eBook ISBN:  978-1-4704-5190-5
Product Code:  AMSTEXT/35.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Sale Price: $55.25
Hardcover ISBN:  978-1-4704-5016-8
eBook ISBN:  978-1-4704-5190-5
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
Sale Price: $119.60 $91.98
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 352019; 488 pp
    MSC: 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 junior-senior level.

    Readership

    Undergraduate 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 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
    • Back Cover
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 352019; 488 pp
MSC: 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 junior-senior level.

Readership

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 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
  • 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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