Overview

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.

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<p>From the reviews: <p> Harold Edwards is well known for his books with a constructivist slant, and his latest book aims to spread his message further. The major part of the book consists of essays telling a connected story, showing what can be achieved with such a constructive restriction imposed. The achievement is impressive . (John Baylis, The Mathematical Gazette, Vol. 90 (5l9), 2006)<p> It is not a book about the history/philosophy of mathematics but rather a very serious book of mathematics. the mathematics is accessible to those with advanced undergraduate or graduate level courses in algebra . Without a doubt the mathematics in this book is rigorous . One of the nice features of this book is the bibliography which notes the sections where each reference appears. It should appeal to mathematicians and historians of mathematics alike. (Bonnie Shulman, MathDL, January, 2005)<p> The author of this volume points out immediately that it is not about the history or philosophy of

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