Essays in Constructive Mathematics

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