Contents and treatment are fresh and very different from the standard treatments
Presents a fully constructive version of what it means to do algebra
The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader
Author(s): Harold M. Edwards
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Pages: 324
City: Cham
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
