Essays in constructive mathematics / / Harold M. Edwards

• Record Nr.: UNISA996490345103316
• Autore: Edwards Harold M.
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 3-030-98558-X
• Edizione: [Second edition.]
• Descrizione fisica: 1 online resource (325 pages)
• Disciplina: 511.3
• Soggetti: Constructive mathematics; Matemàtica constructiva; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- To the Reader -- Contents -- Part I The Original Essays -- 1 A Fundamental Theorem -- Essay 1.1 General Arithmetic -- Essay 1.2 A Fundamental Theorem -- Essay 1.3 Root Fields (Simple Algebraic Extensions) -- Essay 1.4 Factorization of Polynomials with Integer Coefficients -- Essay 1.5 A Factorization Algorithm -- Essay 1.6 Validation of the Factorization Algorithm -- Essay 1.7 About the Factorization Algorithm -- Essay 1.8 Proof of the Fundamental Theorem -- Essay 1.9 Minimal Splitting Polynomials -- 2 Topics in Algebra -- Essay 2.1 Galois's Fundamental Theorem -- Essay 2.2 Algebraic Quantities -- Essay 2.3 Adjunctions and the Factorization of Polynomials -- Essay 2.4 The Splitting Field of xⁿ + c₁xⁿ⁻¹ + c₂xⁿ⁻² + · · · + cₙ -- Essay 2.5 A Fundamental Theorem of Divisor Theory -- 3 Some Quadratic Problems -- Essay 3.1 The Problem A□ + B = □ and "Hypernumbers -- Essay 3.2 Modules -- Essay 3.3 The Class Semigroup. Solution of A□ + B = □ -- Essay 3.4 Multiplication of Modules and Module Classes -- Essay 3.5 Is A a Square Mod p? -- Essay 3.6 Gauss's Composition of Forms -- Essay 3.7 The Construction of Compositions -- 4 The Genus of an Algebraic Curve -- Essay 4.1 Abel's Memoir -- Essay 4.2 Euler's Addition Formula -- Essay 4.3 An Algebraic Definition of the Genus -- Essay 4.4 Newton's Polygon -- Essay 4.5 Determination of the Genus -- Essay 4.6 ; Holomorphic Differentials -- Essay 4.7 The Riemann-Roch Theorem -- Essay 4.8 The Genus Is a Birational Invariant -- 5 Miscellany -- Essay 5.1 On the So-Called Fundamental Theorem of Algebra -- Essay 5.2 Proof by Contradiction and the Sylow Theorems -- Essay 5.3 Overview of ``Linear Algebra'' -- Essay 5.4 The Spectral Theorem -- Essay 5.5 Kronecker as One of E. T. Bell's ``Men of Mathematics'' -- Part II Three Topics in Rational Arithmetic.; 6 Constructive Algebra -- Essay 6.1 Fields -- Essay 6.2 Factorization -- Essay 6.3 Algorithms -- Endnotes to Chapter 6 -- 7 The Algorithmic Foundations of Galois's Theory -- Essay 7.1 What Are the Roots of a Polynomial? -- Essay 7.2 Galois's Overlooked Construction -- Essay 7.3 The Idea of Galois's Construction -- Essay 7.4 Galois's Determination of the Roots of f(x) in Q(V) -- Essay 7.5 The Galois Group and Galois's Proposition 1 -- Essay 7.6 The Foundation -- Essay 7.7 Extension of the Notion of the Galois Group of a Polynomial -- Essay 7.8 The Solution of Cubic Polynomials -- Essay 7.9 Galois's Proposition 2 -- Essay 7.10 Galois's Proposition 3 -- Essay 7.11 Solvability by Radicals -- Essay 7.12 The Simplest Case -- Essay 7.13 Solvability by Radicals Implies Solvability of the Galois Group -- Essay 7.14 Roots of Unity -- Essay 7.15 Solvability of the Galois Group implies Solvability by Radicals -- Essay 7.16 The Explicit Solution of Cubics by Radicals -- 8 A Constructive Definition of Points on an Algebraic Curve -- Essay 8.1 The Points on an Algebraic Curve -- Essay 8.2 Curve Fields That Contain Indeterminates -- Essay 8.3 Points of Curve Fields of the Form A(x) -- Essay 8.4 Expansions at a Point Where x Is a Local Parameter -- Essay 8.5 Augmentation and the Newton Diagram -- Essay 8.6 Stub Solutions and Ambiguities -- Essay 8.7 The Algorithm of Essay 8.6 Must Terminate -- Essay 8.8 Expansion Rules -- Essay 8.9 Points -- Essay 8.10 Points Are Intrinsic to K -- Essay 8.11 Quantities Integral Over x -- Essay 8.12 Points as Divisors -- 9 Abel's Theorem -- Essay 9.1 What Was Abel's Theorem? -- Essay 9.2 Normal Bases -- Essay 9.3 The Field of Constants -- Essay 9.4 Differentials and Holomorphic Differentials -- Essay 9.5 The Construction of Holomorphic Differentials.; Essay 9.6 Parametric Points Constructed Using a Normal Basis -- Essay 9.7 The Theorem of Abel's Last Paper -- Essay 9.8 A Theorem About Holomorphic Differentials -- Essay 9.9 A Change of Parameters -- Essay 9.10 Abel's Addition Theorems -- Essay 9.11 Addition on an Elliptic Curve -- Endnotes to Chapter 9 -- Postscript -- References -- Index.