Essays in Constructive Mathematics

Constructive mathematics is promoted through essays that practice finite algorithms, covering topics from algebra, number theory, and algebraic curves. The second edition adds new essays that expand upon the first, with topics derived from classic works of nineteenth-century mathematics. The essays require mathematical maturity and some prior knowledge of Galois theory, but experience with constructive mathematics is not necessary. Readers should be willing to set aside abstract notions of infinity and explore deep mathematics via explicit constructions.

Format: Hardback
Length: 322 pages
Publication date: 30 September 2022
Publisher: Springer Nature Switzerland AG

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 Galoiss theory of algebraic equations, Gausss theory of binary quadratic forms, and Abels 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 new essays that give deeper and more complete accounts of Galoiss theory, points on an algebraic curve, and Abels theorem. Readers will experience the full power of Galoiss approach to solvability by radicals, learn how to construct points on an algebraic curve using Newtons 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.

Weight: 670g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030985578
Edition number: 2nd ed. 2022