Advanced number theory with applications

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-functionIntroduction to p-Adic AnalysisSolving Modulo pn Introduction to Valuations Non-Archimedean vs. Archimedean Valuations Representation of p-Adic NumbersDirichlet: Characters, Density, and Primes in Progression Dirichlet Characters Dirichlet's L-Function and Theorem Dirichlet DensityApplications to Diophantine Equations Lucas-Lehmer Theory Generalized Ramanujan-Nagell Equations Bachet's Equation The Fermat Equation Catalan and the ABC-Conjecture Elliptic CurvesThe 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 FLTAppendix: Sieve Methods Bibliography Solutions to Odd-Numbered Exercises Index: List of SymbolsIndex: Alphabetical Listing.