Overview
- Introduces new insights into Galois theory, number theory, and algebraic curves from a leading expert
- Offers constructive formulations of Galois theory and the theory of algebraic curves
- Invites readers new to constructive methods to explore deep mathematics through a constructive lens
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Table of contents (9 chapters)
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Front Matter
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The Original Essays
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Front Matter
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Three Topics in Rational Arithmetic
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Front Matter
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Back Matter
Keywords
- constructive mathematics
- constructive approach to mathematical proof
- Kronecker general arithmetic
- Abel's theorem constructive formulation
- constructive algebra
- Galois theory constructive approach
- algorithmic foundations of Galois theory
- constructive definition of points on an algebraic curve
- genus of an algebraic curve
- Newton's diagram
- Gauss binary quadratic forms
About this book
This collection of essays aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it. All definitions and proofs are based on finite algorithms, which pave illuminating paths to nontrivial results, primarily in algebra, number theory, and the theory of algebraic curves. The second edition adds a new set of essays that reflect and expand upon the first.
The topics covered derive from classic works of nineteenth-century mathematics, among them Galois’s theory of algebraic equations, Gauss’s theory of binary quadratic forms, and Abel’s theorems about integrals of rational differentials on algebraic curves. Other topics include Newton's diagram, the fundamental theorem of algebra, factorization of polynomials over constructive fields, and the spectral theorem for symmetric matrices, all treated using constructive methods in the spirit of Kronecker.
In this second edition, the essays of the first edition are augmented with newessays that give deeper and more complete accounts of Galois’s theory, points on an algebraic curve, and Abel’s theorem. Readers will experience the full power of Galois’s approach to solvability by radicals, learn how to construct points on an algebraic curve using Newton’s diagram, and appreciate the amazing ideas introduced by Abel in his 1826 Paris memoir on transcendental functions.Mathematical maturity is required of the reader, and some prior knowledge of Galois theory is helpful. But experience with constructive mathematics is not necessary; readers should simply be willing to set aside abstract notions of infinity and explore deep mathematics via explicit constructions.
Authors and Affiliations
About the author
Harold M. Edwards [1936–2020] was Professor Emeritus of Mathematics at New York University. His research interests lay in number theory, algebra, and the history and philosophy of mathematics. He authored numerous books, including Riemann’s Zeta Function (1974, 2001) and Fermat’s Last Theorem (1977), for which he received the Leroy P. Steele Prize for mathematical exposition in 1980.
David A. Cox (Contributing Author) is Professor Emeritus of Mathematics in the Department of Mathematics and Statistics of Amherst College. He received the Leroy P. Steele Prize for mathematical exposition in 2016 for his book Ideals, Varieties, and Algorithms, with John Little and Donal O’Shea.
Bibliographic Information
Book Title: Essays in Constructive Mathematics
Authors: Harold M. Edwards
DOI: https://doi.org/10.1007/978-3-030-98558-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-98557-8Published: 30 September 2022
Softcover ISBN: 978-3-030-98560-8Published: 02 October 2023
eBook ISBN: 978-3-030-98558-5Published: 29 September 2022
Edition Number: 2
Number of Pages: XIV, 322
Number of Illustrations: 65 b/w illustrations, 325 illustrations in colour
Topics: Mathematics, general, History of Mathematical Sciences, Mathematical Logic and Foundations, Field Theory and Polynomials, Algebraic Geometry