Essays in constructive mathematics建设性数学论文
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作者： Harold M. Edwards 著
出 版 社：
出版时间： 2004-11-1字数：版次： 1页数： 211印刷时间： 2004/11/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387219783包装： 精装内容简介
This book aims to promote constructive mathematics not by defining it or formalizing it but by practicing it. This means that its definitions and proofs use finite algorithms, not `algorithms' that require surveying an infinite number of possibilities to determine whether a given condition is met.
The topics covered derive from classic works of nineteenth century mathematics---among them Galois' theory of algebraic equations, Gauss's theory of binary quadratic forms and Abel's theorem about integrals of rational differentials on algebraic curves. It is not surprising that the first two topics can be treated constructively---although the constructive treatments shed a surprising amount of light on them---but the last topic, involving integrals and differentials as it does, might seem to call for infinite processes. In this case too, however, finite algorithms suffice to define the genus of an algebraic curve, to prove that birationally equivalent curves have the same genus, and to prove the Riemann-Roch theorem. The main algorithm in this case is Newton's polygon, which is given a full treatment. Other topics covered include the fundamental theorem of algebra, the factorization of polynomials over an algebraic number field, and the spectral theorem for symmetric matrices.
作者简介：
Harold M. Edwards is Emeritus Professor of Mathematics at New York University. His previous books are Advanced Calculus (1969, 1980, 1993), Riemann's Zeta Function (1974, 2001), Fermat's Last Theorem (1977), Galois Theory (1984), Divisor Theory (1990) and Linear Algebra (1995). Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new.
目录
Preface
Synopsis
1 A Fundamental Theorem
1.1 General Arithmetic
1.2 A Fundamental Theorem
1.3 Root Fields (Simple Algebraic Extensions)
1.4 Factorization of Polynomials with Integer Coefficients ...
1.5 A Factorization Algorithm
1.6 Validation of the Factorization Algorithm
1.7 About the Factorizati0n Algorithm
1.8 Proof of the Fundamental Theorem
1.9 Minimal Splitting Polynomials
2 Topics in Algebra
2.1 Galois's Fundamental Theorem
2.2 Algebraic Quantities
2.3 Adjunctions and the Factorization of Polynomials
2.4 The Splitting Field of xn+c1xn-1+c2xn-2+...+cn
2.5 A Pundamental Theorem of Divisor Theory
3 Some Quadratic Problems
3.1 The Problem An + B = [] and "Hypernumbers" .
3.2 Modules
3.3 The Class Semigroup. Solution of A[] + B = []
3.4 Multiplication of Modules and Module Classes
3.5 Is A a Square Mod p?
3.6 Gauss's Composition of Forms
3.7 The Construction of Compositions
4 The Genus of an Algebraic Curve
4.1 Abel's Memoir
4.2 Euler's Addition Formula
4.3 An Algebraic Definition of the Genus
4.4 Newton's Polygon
4.5 Determination of the Genus
4,6 Holomorphic Differentials
4.7 The Riemann-Roch Theorem
4.8 The Cenus Is a Birational Invariant
5 Miscellany
5.1 On the So-Called Fundainental Theorem of Algebra
5.2 Proof by Contradiction and the Sylow Theorems
5.3 Overview of 'Linear Algebra'
5.4 The Spectral Theorem
5.5 Kronecker as One of E. T. Bell's "Men of Mathematics"
References
Index
