二次无理数:经典数论入门:an introduction to classical number theory:英文
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作者:
[德]弗朗兹·霍尔特-科赫
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出版社:
哈尔滨工业大学出版社有限公司
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ISBN:
9787560399720
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出版时间:
2021-07
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装帧:
平装
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开本:
16开
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作者:
[德]弗朗兹·霍尔特-科赫
-
出版社:
哈尔滨工业大学出版社有限公司
-
ISBN:
9787560399720
-
出版时间:
2021-07
售价
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83.46
6.0折
定价
￥138.00
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上书时间2024-04-09
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目录
Foreword
Introduction and Preface to the Reader
Notations
Chapter 1 Quadratic irrationals
1.1 Quadratic irrationals, quadratic number fields and discriminants
1.2 The modular group
1.3 Reduced quadratic irrationals
1.4 Two short tables of class numbers
Chapter 2 Continued fractions
2.1 General theory of continued fractions
2.2 Continued fractions of quadratic irrationals Ⅰ: General theory
2.3 Continued fractions of quadratic irrationals Ⅱ: Spe types
Chapter 3 Quadratic residues and Gauss sums
3.1 Elementary theory of power residues
3.2 Gauss and Jacobi sumsThe quadratic reciprocity law
3.4 Sums of two squares
3.5 Kronecker and quadratic symbols
Chapter 4 L-series and Dirichlets prime number theorem
4.1 Preliminaries and some elementary cases
4.2 Multiplicative functions
4.3 Dirichlet L-functions and proof of Dirichlets theorem
4.4 Summation of L-series
Chapter 5 Quadratic orders
5.1 Lattices and orders in quadratic number fields
5.2 Units in quadratic orders
5.3 Lattices and (invertible) fractional ideals in quadratic orders
5.4 Structure of ideals in quadratic orders
5.5 Class groups and class semigroups
5.6 Ambiguous ideals and ideal classes
5.7 An application: Some binary Diophantine equations
5.8 Prime ideals and multiplicative ideal theory
5.9 Class groups of quadratic orders
Chapter 6 Binary quadratic forms
6.1 Elementary definitions and equivalence relations
6.2 Representation of integers
6.3 Reduction Composition
6.5 Theory of genera
6.6 Ternary quadratíc forms
6.7 Sums of squares
Chapter 7 Cubic and biquadratic residues
7.1 The cubic Jacobi symbol
7.2 The cubic reciprocity law
7.3 The biquadratic Jacobi symbol
7.4 The biquadratic reciprocity law
7.5 Rational biquadratic reciprocity laws
7.6 A biquadratic class group character and applications
Chapter 8 Class groups
8.1 The analytic class number formula
8.2 L-functions of quadratic orders
8.3 Amblguous classes and classes of order divisibility by
8.4 Discriminants with cyclic 2-class group: Divisibility by 8 and 16
Appendix A Review of elementary algebra and number theory
A.1 Fundamentals of group theory
A.2 Fundamentals of ring theory
A.3 Elementary arithmetic in Z
A.4 Lattices
A.5 Finite abelian groups
A.6 Prime residue class groups
A.7 Roots of unity and characters of finite abelian groups
A.8 Factorization in integral domains
A.9 Algebraic integers
Appendix B Some results from analysis
B.1 Notational conventions and results from complex analysis
B.2 Further analytic tools
Bibliography
List of Symbols
Subject Index
编辑手记
内容摘要
书是版权引进自泰勒公司的英文原版教科书。本书对经典的二次无理数论给出了统一处理.材料以一种现代和初等代数的安排形式呈现，作者着重介绍了等价、连分数、二次特征标、二次阶、二元二次型和类群本书强调了二元形式的高斯理论与二次阶算法之间的联系，汇集了此前难以理解的和散落在文献中的理论的基本结果，包括二元二次丢番图方程、显式连分数、四次类群特征标、类数的16可除性、F.梅尔滕斯（F,Mertens)对高斯加倍定理的证明，以及一个脱离了基本判别式的二元二次型理论.本书还证明了等差数列中素数的迪利克雷定理，涵盖了迪利克雷的类数公式，证明了每一个初等二元二次形式表示无穷多个素数.在附录中给出了学习代数和初等数论所必需的基本知识。
精彩内容
本书作者为弗朗兹·霍尔特-科赫（Franz Halter-Koch）,他是奥地利格拉茨大学的数学教授。 《二次无理数：经典数论入门》对经典的二次无理数论给出了统一处理。材料以一种现代和初等代数的安排形式呈现，作者着重介绍了等价、连分数、二次特征标、二次阶、二元二次型和类群。 本书强调了二元形式的高斯理论与二次阶算法之间的联系，汇集了此前难以理解的和散落在文献中的理论的基本结果，包括二元二次丢番图方程、显式连分数、四次类群特征标、类数的16可除性、F.梅尔滕斯（F.Mertens）对高斯加倍定理的证明，以及一个脱离了基本判别式的二元二次型理论，本书还证明了等差数列中素数的迪利克雷定理，涵盖了迪利克雷的类数公式，证明了每一个初等二元二次形式表示无穷多个素数。在附录中给出了学习代数和初等数论所必需的基本知识。
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