Advanced Number Theory with Applications Book

Algebraic Number Theory and Quadratic Fields

Algebraic Number Fields

The Gaussian Field

Euclidean Quadratic Fields

Applications of Unique Factorization

Ideals

The Arithmetic of Ideals in Quadratic Fields

Dedekind Domains

Application to Factoring

Binary Quadratic Forms

Basics

Composition and the Form Class Group

Applications via Ambiguity

Genus

Representation

Equivalence Modulo p

Diophantine Approximation

Algebraic and Transcendental Numbers

Transcendence

Minkowski’s Convex Body Theorem

Arithmetic Functions

The Euler–Maclaurin Summation Formula

Average Orders

The Riemann zeta-function

Introduction to p-Adic Analysis

Solving Modulo pⁿ

Introduction to Valuations

Non-Archimedean vs. Archimedean Valuations

Representation of p-Adic Numbers

Dirichlet: Characters, Density, and Primes in Progression

Dirichlet Characters

Dirichlet’s L-Function and Theorem

Dirichlet Density

Applications to Diophantine Equations

Lucas–Lehmer Theory

Generalized Ramanujan–Nagell Equations

Bachet’s Equation

The Fermat Equation

Catalan and the ABC-Conjecture

Elliptic Curves

The Basics

Mazur, Siegel, and Reduction

Applications: Factoring and Primality Testing

Elliptic Curve Cryptography (ECC)

Modular Forms

The Modular Group

Modular Forms and Functions

Applications to Elliptic Curves

Shimura–Taniyama–Weil and FLT

Appendix: Sieve Methods

Bibliography

Solutions to Odd-Numbered Exercises

Index: List of Symbols

Index: Alphabetical Listing