Number theory and geometry

• Verfasst von: Lozano-Robledo, Álvaro [VerfasserIn]
• Titel: Number theory and geometry
• Titelzusatz: an introduction to arithmetic geometry
• Verf.angabe: Álvaro Lozano-Robledo
• Verlagsort: Providence, Rhode Island
• Verlag: American Mathematical Society
• E-Jahr: 2019
• Jahr: [2019]
• Umfang: xv, 488 Seiten
• Illustrationen: Illustrationen, Diagramme
• Gesamttitel/Reihe: Pure and applied undergraduate texts ; 35
• Fussnoten: Literaturverzeichnis: Seite 479-482
• ISBN: 978-1-4704-5016-8
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Online-AusgabeLozano-Robledo, Álvaro: Number Theory and Geometry. - Providence : American Mathematical Society, 2019. - 1 online resource (506 pages)
• RVK-Notation: SK 240
• RVK-Notation: SK 180
• K10plus-PPN: 166408326X

Abstract

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.

Abstract

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/ \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.1130