Cargando…
Discriminant equations in Diophantine number theory /
The first comprehensive and up-to-date account of discriminant equations and their applications. For graduate students and researchers.
| Clasificación: | Libro Electrónico |
|---|---|
| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge, UK :
Cambridge University Press,
2016.
|
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Half title; Series; Title; Copyright; Contents; Preface; Acknowledgments; Summary; Part One Preliminaries; 1 Finite Étale Algebras over Fields; 1.1 Terminology for Rings and Algebras; 1.2 Finite Field Extensions; 1.3 Basic Facts on Finite Étale Algebras over Fields; 1.4 Resultants and Discriminants of Polynomials; 1.5 Characteristic Polynomial, Trace, Norm, Discriminant; 1.6 Integral Elements and Orders; 2 Dedekind Domains; 2.1 Definitions; 2.2 Ideal Theory of Dedekind Domains; 2.3 Discrete Valuations; 2.4 Localization; 2.5 Integral Closure in Finite Field Extensions
- 2.6 Extensions of Discrete Valuations2.7 Norms of Ideals; 2.8 Discriminant and Different; 2.9 Lattices over Dedekind Domains; 2.10 Discriminants of Lattices of Étale Algebras; 3 Algebraic Number Fields; 3.1 Definitions and Basic Results; 3.1.1 Absolute Norm of an Ideal; 3.1.2 Discriminant, Class Number, Unit Group and Regulator; 3.1.3 Explicit Estimates; 3.2 Absolute Values: Generalities; 3.3 Absolute Values and Places on Number Fields; 3.4 S-integers, S-units and S-norm; 3.5 Heights and Houses; 3.6 Estimates for Units and S-units
- 3.7 Effective Computations in Number Fields and Étale Algebras3.7.1 Algebraic Number Fields; 3.7.2 Relative Extensions and Finite Étale Algebras; 4 Tools from the Theory of Unit Equations; 4.1 Effective Results over Number Fields; 4.1.1 Equations in Units of Rings of Integers; 4.1.2 Equations with Unknowns from a Finitely Generated Multiplicative Group; 4.2 Effective Results over Finitely Generated Domains; 4.3 Ineffective Results, Bounds for the Number of Solutions; Part Two Monic Polynomials and Integral Elements of Given Discriminant, Monogenic Orders; 5 Basic Finiteness Theorems
- 5.1 Basic Facts on Finitely Generated Domains5.2 Discriminant Forms and Index Forms; 5.3 Monogenic Orders, Power Bases, Indices; 5.4 Finiteness Results; 5.4.1 Discriminant Equations for Monic Polynomials; 5.4.2 Discriminant Equations for Integral Elements in Étale Algebras; 5.4.3 Discriminant Form and Index Form Equations; 5.4.4 Consequences for Monogenic Orders; 6 Effective Results over Z; 6.1 Discriminant Form and Index Form Equations; 6.2 Applications to Integers in a Number Field; 6.3 Proofs; 6.4 Algebraic Integers of Arbitrary Degree; 6.5 Proofs
- 6.6 Monic Polynomials of Given Discriminant6.7 Proofs; 6.8 Notes; 6.8.1 Some Related Results; 6.8.2 Generalizations over Z; 6.8.3 Other Applications; 7 Algorithmic Resolution of Discriminant Form and Index Form Equations; 7.1 Solving Discriminant Form and Index Form Equations via Unit Equations, A General Approach; 7.1.1 Quintic Number Fields; 7.1.2 Examples; 7.2 Solving Discriminant Form and Index Form Equations via Thue Equations; 7.2.1 Cubic Number Fields; 7.2.2 Quartic Number Fields; 7.2.3 Examples; 7.3 The Solvability of Index Equations in Various Special Number Fields; 7.4 Notes