Advanced Topics in Computational Number Theory (häftad)
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Format
Häftad (Paperback / softback)
Språk
Engelska
Antal sidor
581
Utgivningsdatum
2012-10-13
Upplaga
Softcover reprint of the original 1st ed. 2000
Förlag
Springer-Verlag New York Inc.
Illustrationer
XV, 581 p.
Dimensioner
234 x 156 x 30 mm
Vikt
830 g
Antal komponenter
1
Komponenter
1 Paperback / softback
ISBN
9781461264194
Advanced Topics in Computational Number Theory (häftad)

Advanced Topics in Computational Number Theory

Häftad Engelska, 2012-10-13
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Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.
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"Das vorliegende Buch ist eine Fortsetzung des bekannten erkes "A Course in Computational Algebraic Number Theory" (Graduate Texts in Mathematics 138) desselben Autors. ... So ist das vorliegende Buch ein sehr umfangliches Nachschlagewerk zur algorithmischen Zahlentheorie, das zusammen mit dem ersten Buch des Autors sicherlich eine Standard-Referenz fur zahlentheoretische Algorithmen darstellen wird." Internationale Mathematische Nachrichten, Nr. 187, August 2001

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Innehållsförteckning

1. Fundamental Results and Algorithms in Dedekind Domains.- 1.1 Introduction.- 1.2 Finitely Generated Modules Over Dedekind Domains.- 1.2.1 Finitely Generated Torsion-Free and Projective Modules.- 1.2.2 Torsion Modules.- 1.3 Basic Algorithms in Dedekind Domains.- 1.3.1 Extended Euclidean Algorithms in Dedekind Domains.- 1.3.2 Deterministic Algorithms for the Approximation Theorem.- 1.3.3 Probabilistic Algorithms.- 1.4 The Hermite Normal Form Algorithm in Dedekind Domains.- 1.4.1 Pseudo-Objects.- 1.4.2 The Hermite Normal Form in Dedekind Domains.- 1.4.3 Reduction Modulo an Ideal.- 1.5 Applications of the HNF Algorithm.- 1.5.1 Modifications to the HNF Pseudo-Basis.- 1.5.2 Operations on Modules and Maps.- 1.5.3 Reduction Modulo p of a Pseudo-Basis.- 1.6 The Modular HNF Algorithm in Dedekind Domains.- 1.6.1 Introduction.- 1.6.2 The Modular HNF Algorithm.- 1.6.3 Computing the Transformation Matrix.- 1.7 The Smith Normal Form Algorithm in Dedekind Domains.- 1.8 Exercises for Chapter 1.- 2. Basic Relative Number Field Algorithms.- 2.1 Compositum of Number Fields and Relative and Absolute Equations.- 2.1.1 Introduction.- 2.1.2 Etale Algebras.- 2.1.3 Compositum of Two Number Fields.- 2.1.4 Computing (?1 and ?2.- 2.1.5 Relative and Absolute Defining Polynomials.- 2.1.6 Compositum with Normal Extensions.- 2.2 Arithmetic of Relative Extensions.- 2.2.1 Relative Signatures.- 2.2.2 Relative Norm, Trace, and Characteristic Polynomial.- 2.2.3 Integral Pseudo-Bases.- 2.2.4 Discriminants.- 2.2.5 Norms of Ideals in Relative Extensions.- 2.3 Representation and Operations on Ideals.- 2.3.1 Representation of Ideals.- 2.3.2 Representation of Prime Ideals.- 2.3.3 Computing Valuations.- 2.3.4 Operations on Ideals.- 2.3.5 Ideal Factorization and Ideal Lists.- 2.4 The Relative Round 2 Algorithm and Related Algorithms.- 2.4.1 The Relative Round 2 Algorithm.- 2.4.2 Relative Polynomial Reduction.- 2.4.3 Prime Ideal Decomposition.- 2.5 Relative and Absolute Representations.- 2.5.1 Relative and Absolute Discriminants.- 2.5.2 Relative and Absolute Bases.- 2.5.3 Ups and Downs for Ideals.- 2.6 Relative Quadratic Extensions and Quadratic Forms.- 2.6.1 Integral Pseudo-Basis, Discriminant.- 2.6.2 Representation of Ideals.- 2.6.3 Representation of Prime Ideals.- 2.6.4 Composition of Pseudo-Quadratic Forms.- 2.6.5 Reduction of Pseudo-Quadratic Forms.- 2.7 Exercises for Chapter 2.- 3. The Fundamental Theorems of Global Class Field Theory.- 3.1 Prologue: Hilbert Class Fields.- 3.2 Ray Class Groups.- 3.2.1 Basic Definitions and Notation.- 3.3 Congruence Subgroups: One Side of Class Field Theory.- 3.3.1 Motivation for the Equivalence Relation.- 3.3.2 Study of the Equivalence Relation.- 3.3.3 Characters of Congruence Subgroups.- 3.3.4 Conditions on the Conductor and Examples.- 3.4 Abelian Extensions: The Other Side of Class Field Theory.- 3.4.1 The Conductor of an Abelian Extension.- 3.4.2 The Frobenius Homomorphism.- 3.4.3 The Artin Map and the Artin Group Am(L/K).- 3.4.4 The Norm Group (or Takagi Group) Tm(L/K).- 3.5 Putting Both Sides Together: The Takagi Existence Theorem 154.- 3.5.1 The Takagi Existence Theorem.- 3.5.2 Signatures, Characters, and Discriminants.- 3.6 Exercises for Chapter 3.- 4. Computational Class Field Theory.- 4.1 Algorithms on Finite Abelian groups.- 4.1.1 Algorithmic Representation of Groups.- 4.1.2 Algorithmic Representation of Subgroups.- 4.1.3 Computing Quotients.- 4.1.4 Computing Group Extensions.- 4.1.5 Right Four-Term Exact Sequences.- 4.1.6 Computing Images, Inverse Images, and Kernels.- 4.1.7 Left Four-Term Exact Sequences.- 4.1.8 Operations on Subgroups.- 4.1.9 p-Sylow Subgroups of Finite Abelian Groups.- 4.1.10 Enumeration of Subgroups.- 4.1.11 Application to the Solution of Linear Equations and Congruences.- 4.2 Computing the Structure of (?K/m)*.- 4.2.1 Standard Reductions of the Problem.- 4.2.2 The Use of p-adic Logarithms.- 4.2.3 Computing (?K/pk)* by Induction.- 4.2.4 Representation of Elements of (?K/m)*.- 4.2.5 Computing (?K/m)*.