计算代数数论教程(英文版)
作者： (法)科恩
出版时间：2015年7月
出版社：世界图书出版公司
- 世界图书出版公司
- 9787510097973
- 148109
- 2015年7月
- 未分类
- 未分类
- O156.2
由科恩著的《计算代数数论教程(英文版)》介绍了148种算法，它们是数论计算的基础，其中包括与数论、椭圆曲线、素性测定和因式分解等相关的计算。 书中对每种算法都作了完整的理论介绍，将学习者需要的理论基础降到最低。书中对每个算法的详细描述实现了其直接在计算机上的运行，并且给出了众多的进一步的执行提示。书中的许多算法在别的书上从来没有被看到过，或者说它们第一次以书的形式出现在我们面前。
Chapter 1 Fundamental Number-Theoretic Algorithms
1.1 Introduction
1.1.1 Algorithms
1.1.2 Multi-precision
1.1.3 Base Fields and Rings
1.1.4 Notations
1.2 The Powering Algorithms
1.3 Euclid's Algorithms
1.3.1 Euclid's and Lehmer's Algorithms
1.3.2 Euclid's Extended Algorithms
1.3.3 The Chinese Remainder Theorem
1.3.4 Continued Fraction Expansions of Real Numbers
1.4 The Legendre Symbol
1.4.1 The Groups (Z/nZ)*
1.4.2 The Legendre-Jacobi-Kronecker Symbol
1.5 Computing Square Roots Modulo p
1.5.1 The Algorithm of ToneUi and Shanks
1.5.2 The Algorithm of Cornacchia
1.6 Solving Polynomial Equations Modulo p
1.7 Power Detection
1.7.1 Integer Square Roots
1.7.2 Square Detection
1.7.3 Prime Power Detection
1.8 Exercises for Chapter 1
Chapter 2 Algorithms for Linear Algebra and Lattices
2.1 Introduction
2.2 Linear Algebra Algorithms on Square Matrices
2.2.1 Generalities on Linear Algebra Algorithms
2.2.2 Gaussian Elimination and Solving Linear Systems
2.2.3 Computing Determinants
2.2.4 Computing the Characteristic Polynomial
2.3 Linear Algebra on General Matrices
2.3.1 Kernel and Image
2.3.2 Inverse Image and Supplement
2.3.3 Operations on Subspaces
2.3.4 Remarks on Modules
2.4 Z-Modules and the Hermite and Smith Normal Forms
2.4.1 Introduction to Z-Modules
2.4.2 The Hermite Normal Form
2.4.3 Applications of the Hermite Normal Form
2.4.4 The Smith Normal Form and Applications
2.5 Generalities on Lattices
2.5.1 Lattices and Quadratic Forms
2.5.2 The Gram-Schmidt Orthogonalization Procedure
2.{} Lattice Reduction Algorithms
2.6.1 The LLL Algorithm
2.6.2 The LLL Algorithm with Deep Insertions
2.6.3 The Integral LLL Algorithm
2.6.4 LLL Algorithms for Linearly Dependent Vectors
2.7 Applications of the LLL Algorithm
2.7.1 Computing the Integer Kernel and Image of a Matrix
2.7.2 Linear and Algebraic Dependence Using LLL
2.7.3 Finding Small Vectors in Lattices
2.8 Exercises for Chapter 2
Chapter 3 Algorithms on Polynomials
3.1 Basic Algorithms
3.1.1 Representation of Polynomials
3.1.2 Multiplication of Polynomials
3.1.3 Division of Polynomials
3.2 Euclid's Algorithms for Polynomials
3.2.1 Polynomials over a Field
3.2.2 Unique Factorization Domains (UFD's)
3.2.3 Polynomials over Unique Factorization Domains
……
Chapter 4 Algorithms for Algebraic Number Theory Ⅰ
Chapter 5 Algorithms for Quadratic Fields
Chapter 6 Algorithms—for Algebraic Number Theory Ⅱ
Chapter 7 Introduction to Elliptic Curves
Chapter 8 Factoring in the Dark Ages
Chapter 9 Modern Primality Tests
Chapter 10 Modern Factoring Methods
Appendix A Packages for Number Theory
Appendix B Some Useful Tables
Bibliography
Index
Errata et Addenda
