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

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This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction.

Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach.

Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.

Author(s): Sidney A. Morris, Arthur Jones, Kenneth R. Pearson
Series: Undergraduate Texts in Mathematics: Readings in Mathematics
Edition: 2
Publisher: Springer
Year: 2022

Language: English
Pages: 232
City: Cham

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