Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations

Preface to the Second Edition
Preface to the First Edition
Introduction
	0.1 Four Famous Problems
	0.2 Straightedge and Compass Constructions
	0.3 Impossibility of the Geometric Constructions
	0.4 Solving Polynomial Equations
	Additional Reading for the Introduction
Contents
List of Figures
1 Algebraic Preliminaries
	1.1 Fields, Rings and Vector Spaces
	Exercises 1.1
	1.2 Polynomials
	Exercises 1.2
	1.3 The Division Algorithm
	Exercises 1.3
	1.4 The Rational Roots Test
	Exercises 1.4
	Appendix to Chapter 1
	Additional Reading for Chapter 1
2 Algebraic Numbers and Their Polynomials
	2.1 Algebraic Numbers
	Exercises 2.1
	2.2 Monic Polynomials
	Exercises 2.2
	2.3 Monic Polynomials of Least Degree
	Exercises 2.3
	Additional Reading for Chapter 2
3 Extending Fields
	3.1 An Illustration: double struck upper Q left parenthesis StartRoot 2 EndRoot right parenthesismathbbQ(sqrt2)
	Exercises 3.1
	3.2 Construction of double struck upper F left parenthesis alpha right parenthesismathbbF(α)
	Exercises 3.2
	3.3 Iterating the Construction
	Exercises 3.3
	3.4 Towers of Fields
	Exercises 3.4
	Additional Reading for Chapter 3
4 Irreducible Polynomials
	4.1 Irreducible Polynomials
	Exercises 4.1
	4.2 Reducible Polynomials and Zeros
	Exercises 4.2
	4.3 Irreducibility and irrleft parenthesis alpha comma double struck upper F right parenthesis(α, mathbbF)
	Exercises 4.3
	4.4 Finite-dimensional Extensions
	Exercises 4.4
	Additional Reading for Chapter 4
5 Straightedge and Compass Constructions
	5.1 Standard Straightedge and Compass Constructions
		5.1.1 Bisecting a Line Segment
		5.1.2 Transferring a Length (Using a Compass)
		5.1.3 Bisecting an Angle
		5.1.4 Constructing an Angle of 60°
		5.1.5 Constructing an Angle of 90°
		5.1.6 Copying an Angle
		5.1.7 Constructing a Line Parallel to a Given Line
		5.1.8 Use of Compass and Straightedge
	Exercises 5.1
	5.2 Products, Quotients, Square Roots
		5.2.1 Constructing a Product
		5.2.2 Constructing a Quotient
		5.2.3 Constructing Square Roots
	Exercises 5.2
	5.3 Rules for Straightedge and Compass Constructions
		5.3.1 Construction Rules
		5.3.2 Doubling the Cube
		5.3.3 Squaring the Circle
		5.3.4 Trisecting an Angle
	Exercises 5.3
	5.4 Constructible Numbers and Fields
	Exercises 5.4
	Additional Reading for Chapter 5
6 Proofs of the Geometric Impossibilities
	6.1 Non-Constructible Numbers
	Exercises 6.1
	6.2 The Three Geometric Constructions are Impossible
		6.2.1 Problem I - Doubling the cube.
		6.2.2 Problem II - Trisecting an arbitrary angle.
		6.2.3 Problem III - Squaring the circle.
		6.2.4 Other constructions which are impossible.
	Exercises 6.2
	6.3 All Constructibles Come From Square Roots'' Theorem
	Exercises 6.3
	Additional Reading for Chapter 6
7 Zeros of Polynomials of Degrees 2, 3,  and 4
	7.1 Solving Quadratic Equations
	Exercises 7.1
	7.2 Solving Cubic Equations
	Exercises 7.2
	7.3 Solving Quartic Equations
	Exercises 7.3
	Additional Reading for Chapter 7
8 Quintic Equations I: Symmetric Polynomials
	8.1 Brief History of the Quintic Equation: 1683–1826
	8.2 Fundamental Theorem of Algebra
	Exercises 8.2
	8.3 Primitive and Symmetric Polynomials
	Exercises 8.3
	Additional Reading for Chapter 8
9 Quintic Equations II: The Abel–Ruffini Theorem
	9.1 Algebraically Soluble Polynomials
	Exercises 9.1
	9.2 The Number of Real Number Zeros of an Irreducible Polynomial
	Exercises 9.2
	9.3 Kronecker's Theorem and the Abel–Ruffini Theorem
	Exercises 9.3
	Additional Reading for Chapter 9
10 Transcendence of e and piπ
	10.1 Preliminaries
	Exercises 10.1
	10.2 e is Transcendental
	Exercises 10.2
	10.3 piπ is Transcendental – Part 1
	Exercises 10.3
	10.4 Preliminaries on Complex-valued Integrals
	Exercises 10.4
	10.5 piπ is Transcendental – Part 2
	Exercises 10.5
	10.6 Transcendental Number Theory
	Exercises 10.6
	Additional Reading for Chapter 10
11 An Algebraic Postscript
	11.1 The Ring double struck upper F left bracket upper X right bracket Subscript p left parenthesis upper X right parenthesismathbbF[X]p(X)
	Exercises 11.1
	11.2 Division and Reciprocals in double struck upper F left bracket upper X right bracket Subscript p left parenthesis upper X right parenthesismathbbF[X]p(X)
	Exercises 11.2
	11.3 Reciprocals in double struck upper F left parenthesis alpha right parenthesismathbbF(α)
	Exercises 11.3
	Additional Reading for Chapter 11
12 Other Impossibilities: Regular Polygons and Integration in Finite Terms
	12.1 Construction of Regular Polygons
	12.2 Integration in Closed Form
	Exercises 12.2
	Additional Reading for Chapter 12
Appendix  References
	Index