The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development. of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these constructions are impossible. Our approach introduces all the relevant concepts about fields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. We have used this text to teach second-year students at La Trobe University over a period of many years, each time refining the material in the light of student performance.

Inhalt

0.1 Three Famous Problems.- 0.2 Straightedge and Compass Constructions.- 0.3 Impossibility of the Constructions.- 1 Algebraic Preliminaries.- 1.1 Fields, Rings and Vector Spaces.- 1.2 Polynomials.- 1.3 The Division Algorithm.- 1.4 The Rational Roots Test.- Appendix to Chapter 1.- 2 Algebraic Numbers and Their Polynomials.- 2.1 Algebraic Numbers.- 2.2 Monic Polynomials.- 2.3 Monic Polynomials of Least Degree.- 3 Extending Fields.- 3.1 An Illustration: $$\mathbb{Q}(\sqrt 2 )$$.- 3.2 Construction of $$\mathbb{F}(\alpha )$$.- 3.3 Iterating the Construction.- 3.4 Towers of Fields.- 4 Irreducible Polynomials.- 4.1 Irreducible Polynomials.- 4.2 Reducible Polynomials and Zeros.- 4.3 Irreducibility and irr$$(\alpha ,\mathbb{F})$$.- 4.4 Finite-dimensional Extensions.- 5 Straightedge and Compass Constructions.- 5.1 Standard Straightedge and Compass Constructions.- 5.2 Products, Quotients, Square Roots.- 5.3 Rules for Straightedge and Compass Constructions.- 5.4 Constructible Numbers and Fields.- 6 Proofs of the Impossibilities.- 6.1 Non-Constructible Numbers.- 6.2 The Three Constructions are Impossible.- 6.3 Proving the "All Constructibles Come From Square Roots" Theorem.- 7 Transcendence of e and ?.- 7.1 Preliminaries.- 7.2 e is Transcendental.- 7.3 Preliminaries on Symmetric Polynomials.- 7.4 ? is Transcendental - Part 1.- 7.5 Preliminaries on Complex-valued Integrals.- 7.6 ? is Transcendental - Part 2.- 8 An Algebraic Postscript.- 8.1 The Ring $$\mathbb{F}\left[ X \right]{p(X)}$$.- 8.2 Division and Reciprocals in $$\mathbb{F}\left[ X \right]{p(X)}$$.- 8.3 Reciprocals in $$\mathbb{F}\left( \alpha \right)$$.- 9 Other Impossibilities and Abstract Algebra.- 9.1 Construction of Regular Polygons.- 9.2 Solution of Quintic Equations.- 9.3 Integration in Closed Form.