Essays in Constructive Mathematics

by Edwards, Harold M.

• ISBN13: 9780387219783
• ISBN10: 0387219781
• Format: Hardcover
• Copyright: 2004-11-01
• Publisher: Springer Verlag

Summary

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. 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.

Author Biography

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. In 1980 he was awarded the Steele Prize for mathematical exposition for the Riemann and Fermat books.

Table of Contents

Preface	ix
Synopsis	xiii
A Fundamental Theorem	1	(40)
General Arithmetic	1	(5)
A Fundamental Theorem	6	(4)
Root Fields (Simple Algebraic Extensions)	10	(3)
Factorization of Polynomials with Integer Coefficients	13	(7)
A Factorization Algorithm	20	(7)
Validation of the Factorization Algorithm	27	(4)
About the Factorization Algorithm	31	(4)
Proof of the Fundamental Theorem	35	(4)
Minimal Splitting Polynomials	39	(2)
Topics in Algebra	41	(24)
Galois's Fundamental Theorem	41	(5)
Algebraic Quantities	46	(3)
Adjunctions and the Factorization of Polynomials	49	(7)
The Splitting Field of xn + c1xn-1 + c2xn-2 + ... + cn	56	(6)
A Fundamental Theorem of Divisor Theory	62	(3)
Some Quadratic Problems	65	(54)
The Problem A + B and ``Hypernumbers''	65	(6)
Modules	71	(8)
The Class Semigroup. Solution of A + B	79	(14)
Multiplication of Modules and Module Classes	93	(9)
Is A a Square Mod p?	102	(6)
Gauss's Composition of Forms	108	(4)
The Construction of Compositions	112	(7)
The Genus of an Algebraic Curve	119	(60)
Abel's Memoir	119	(5)
Euler's Addition Formula	124	(4)
An Algebraic Definition of the Genus	128	(4)
Newton's Polygon	132	(10)
Determination of the Genus	142	(13)
Holomorphic Differentials	155	(9)
The Riemann-Roch Theorem	164	(7)
The Genus Is a Birational Invariant	171	(8)
Miscellany	179	(26)
On the So-Called Fundamental Theorem of Algebra	179	(7)
Proof by Contradiction and the Sylow Theorems	186	(4)
Overview of `Linear Algebra'	190	(6)
The Spectral Theorem	196	(5)
Kronecker as One of E. T. Bell's ``Men of Mathematics''	201	(4)
References	205	(4)
Index	209