The mathematical content is intensely classical.

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Edwards makes it warmly accessible to any interested reader. And he is breaking fresh ground, in his rigorously constructive or constructivist presentation. So the book will interest anyone trying to learn these major, central topics in classical algebra and algebraic number theory. Also, anyone interested in constructivism, for or against.

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And even anyone who can be intrigued and drawn in by a masterly exposition of beautiful mathematics. This book aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it, by basing all definitions and proofs on finite algorithms. The topics covered derive from classic works of nineteenth century mathematicsamong 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.

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It is not surprising that the first two topics can be treated constructivelyalthough the constructive treatments shed a surprising amount of light on thembut 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.

## Constructivism (philosophy of mathematics)

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. Readers of his Advanced Calculus will know that his preference for constructive mathematics is not new. Skip to main content Skip to table of contents.

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## Essays in Constructive Mathematics

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