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，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. 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.
作者簡介
暫缺《數(shù)論Ⅳ：超越數(shù)（續(xù)一 影印版）》作者簡介
圖書目錄
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
鄂公網(wǎng)安備 42010302001612號(hào) 