| 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》，本书是调查的最重要的研究方向在超越数论。 |
| 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 Liouville's Theorem and Liouville Numbers 1.7 Generalization of Liouville's Theorem 2 Approximations of Algebraic Numbers and Thue's Equation 2.1 Thue's Equation 2.2 The Case n = 2 2.3 The Case n > 3 3 Strengthening Liouville's Theorem First Version of Thue's Method 3.1 A Way to Bound qθ-ρ 3.2 Construction of Rational Approximations for 3.3 Thue's 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 Liouville's Theorem and Thue's Theorem 4.1 The Dirichlet Pigeonhole Principle 4.2 Thue's Method in the General Case 4.3 Thue's Theorem on Approximation of Algebraic Numbers 4.4 The Non-effectiveness of Thue's Theorems 5 Further Development of Thue's Method 5.1 Siegel's Theorem 5.2 The Theorems of Dyson and Gel'fond 5.3 Dyson's Lemma 5.4 Bombieri's Theorem 6 Multidimensional Variants of the Thue-Siegel Method 6.1 Preliminary Remarks 6.2 Siegel's Theorem 6.3 The Theorems of Schneider and Mahler 7 Roth's Theorem 7.1 Statement of the Theorem 7.2 The Index of a Polynomial 7.3 Outline of the Proof of Roth's Theorem 7.4 Approximation of Algebraic Numbers by Algebraic Numbers 7.5 The Number k in Roth's 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 Schmidt's Theorem 8.1 Elementary Estimates 8.2 Schmidt's Theorem 8.3 Minkowski's Theorem on Linear Forms 8.4 Schmidt's Subspace Theorem Chapter 2 Effective Constructions in Transcendental Number Theory Chapter 3 Hillbert's Seventh Problem Chapter 4 Multidimensional Generalization of Hillbert's 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 |
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