数论:Ⅳ:超越数
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作者A.N.Parshin，I.R.Shafarevich[编著]
出版社科学出版社
ISBN9787030235084
出版时间2009-01
装帧精装
开本16开
定价80元
货号4087599
上书时间2025-01-09
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目录
Notation
Introduction
0.1 Preliminary Remarks
0.2 Irrationality of 2
0.3 The Number π
0.4 Transcendental Numbers
0.5 Approximation of Algebraic Numbers
0.6 Transcendence Questions and Other Branches of Number Theory
0.7 The Basic Problems Studied in Transcendental Number Theory
0.8 Different Ways of Giving the Numbers
0.9 Methods
Chapter 1 Approximation of Algebraic Numbers
1 Preliminaries
1.1 Parameters for Algebraic Numbers and Polynomials
1.2 Statement of the Problem
1.3 Approximation of Rational Numbers
1.4 Continued Fractions
1.5 Quadratic Irrationalities
1.6 Liouvilles Theorem and Liouville Numbers
1.7 Generalization of Liouvilles Theorem
2 Approximations of Algebraic Numbers and Thues Equation
2.1 Thues Equation
2.2 The Case n = 2
2.3 The Case n > 3
3 Strengthening Liouvilles Theorem First Version of Thues Method
3.1 A Way to Bound qθ-ρ
3.2 Construction of Rational Approximations for
3.3 Thues First Result
3.4 Effectiveness
3.5 Effective Analogues of Theorem 1.6
3.6 The First Effective Inequalities of Baker
3.7 Effective Bounds on Linear Forms in Algebraic Numbers
4 Stronger and More General Versions of Liouvilles Theorem and Thues Theorem
4.1 The Dirichlet Pigeonhole Principle
4.2 Thues Method in the General Case
4.3 Thues Theorem on Approximation of Algebraic Numbers
4.4 The Non-effectiveness of Thues Theorems
5 Further Development of Thues Method
5.1 Siegels Theorem
5.2 The Theorems of Dyson and Gelfond
5.3 Dysons Lemma
5.4 Bombieris Theorem
6 Multidimensional Variants of the Thue-Siegel Method
6.1 Preliminary Remarks
6.2 Siegels Theorem
6.3 The Theorems of Schneider and Mahler
7 Roths Theorem
7.1 Statement of the Theorem
7.2 The Index of a Polynomial
7.3 Outline of the Proof of Roths Theorem
7.4 Approximation of Algebraic Numbers by Algebraic Numbers
7.5 The Number k in Roths Theorem
7.6 Approximation by Numbers of a Special Type
7.7 Transcendence of Certain Numbers
7.8 The Number of Solutions to the Inequality (62) and Certain Diophantine Equations
8 Linear Forms in Algebraic Numbers and Schmidts Theorem
8.1 Elementary Estimates
8.2 Schmidts Theorem
8.3 Minkowskis Theorem on Linear Forms
8.4 Schmidts Subspace Theorem
Chapter 2 Effective Constructions in Transcendental Number Theory
Chapter 3 Hillberts Seventh Problem
Chapter 4 Multidimensional Generalization of Hillberts Seventh Problem
Chapter 5 Values of Analytic Functions That Satisgy Linear Differential Equations
Chapter 6 Algebraic Independence of the Values of Analytic Functions That Have an Additaon La
内容摘要
This book is a survey of the most important directions of research in transcendental number theory. The central topics in this theory include proofs of irrationality and transcendence of various numbers，espely those，that arise as the values of spe functions. Questions of this sort go back to ancient times. An example is the old problem of squaring the circle，which Lindemann showed to bc impossible in 1882，when hc proved that Pi is a trandental number. Euler's conjecture that the logarithm of an algebraic number to an algebraic base is transcendental was included in Hilbert's famous list of open problems; this conjecture was proved by Gel'fond and Schneider in 1934. A more recent result was Anerv's surprising proof of the irrationality of ξ(3)in 1979.
The quantitative aspects of the theory have important applications to the study of Diophantine equations and other areas of number theory. For a reader interested in different branches of number theory，this monograph provides both an overview of the central ideas and techniques of transcendental number theory，and also a guide to the most important results and references.
主编推荐
本书为《国外数学名著系列》丛书之一。该丛书是科学出版社组织学术界多位知名院士、专家精心筛选出来的一批基础理论类数学著作，读者对象面向数学系高年级本科生、研究生及从事数学专业理论研究的科研工作者。本册为《数论(Ⅳ超越数影印版)65》，本书是调查的很重要的研究方向在超越数论。
精彩内容
《数论4：超越数(影印版)》is a survey of the most important directions of research in transcendental number theory. The central topics in this theory include proofs of irrationality and transcendence of various numbers，especially those，that arise as the values of special functions. Questions of this sort go back to ancient times. An example is the old problem of squaring the circle，which Lindemann showed to bc impossible in 1882，when hc proved that Pi is a trandental number. Eulers conjecture that the logarithm of an algebraic number to an algebraic base is transcendental was included in Hilberts famous list of open problems; this conjecture was proved by Gelfond and Schneider in 1934. A more recent result was Anervs surprising proof of the irrationality of ξ(3)in 1979. The quantitative aspects of the theory have important applications to the study of Diophantine equations and other areas of number theory. For a reader interested in different branches of number theory，this monograph provides both an overview of the central ideas and techniques of transcendental number theory，and also a guide to the most important results and references.
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