Abstract Algebra and Famous Impossibilities: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations (Hardcover)
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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.
About the Author
Sidney A. Morris is Emeritus Professor at the Federation University, Australia (formerly University of Ballarat) and Adjunct Professor at La Trobe University, Australia. His primary research is in topological groups, topology, and transcendental number theory, with broader interests including early detection of muscle wasting diseases, health informatics, and predicting the Australian stock exchange. He is the author of several books. Arthur Jones [1934-2006] and Kenneth R. Pearson [1943-2015] were Professors in Mathematics at La Trobe University, Australia. Each had a great passion for teaching and for mathematics.