عنوان

Number theory and geometry :

147045016X

9781470450168

Number theory and geometry :

[Book]

an introduction to arithmetic geometry /

Álvaro Lozano-Robledo.

Providence, Rhode Island :

American Mathematical Society,

[2019]

xv, 488 pages ;

27 cm.

Pure and applied undergraduate texts ;

35

Includes bibliographical references and index.

Introduction -- Part 1. Integers, Polynomials, Lines, and Congruences: The Integers -- The Prime Numbers Congruences -- Groups, Rings, and Fields -- Finite Fields -- The Theorems of Wilson, Fermat, and Euler -- Primitive Roots -- Part 2. Quadratic Congruences and Quadratic Equations: An Introduction to Quadratic Equations -- Quadratic Congruences -- The Hasse-Minkowski Theorem -- Circles, Ellipses, and the Sum of Two Squares Problem -- Continued Fractions -- Hyperbolas and Pell's Equation -- Part 3. Cubic Equations and Elliptic Curves: An Introduction to Cubic Equations -- Elliptic Curves

0

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. -- Provided by publisher.

Arithmetical algebraic geometry, Textbooks.

Number theory, Textbooks.

Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Arithmetic problems. Diophantine geometry.

Arithmetical algebraic geometry.

Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Arithmetic algebraic geometry (Diophantine geometry).

Number theory.

Number theory.

Lozano-Robledo, Álvaro,1978-

20200823025858.0

rda

[Book]

Y