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

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.

著名的平方圓、倍立方和三等分角度的問題，吸引了專業和業餘數學家們的想像力超過兩千年。儘管這些男性和女性付出了巨大的努力和巧妙的嘗試，這些問題卻無法通過純幾何方法解決。直到19世紀的抽象代數的發展，數學家們才得出了令人驚訝的結論，即這些構造是不可能的。在這本書中，我們發展了足夠的抽象代數來證明這些構造是不可能的。我們的方法以比較具體的方式介紹了有關域的所有相關概念，避免使用商結構（甚至避免使用歐幾里得算法來找到兩個多項式的最大公因數）。將幾何問題作為特定目標，為引入代數概念提供了動機，我們發現學生對此非常積極回應。多年來，我們使用這本教材教授拉特羅布大學的二年級學生，每次根據學生表現不斷完善教材。

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.

Sidney A. Morris是澳洲聯邦大學（前身為巴拉瑞特大學）的名譽教授，也是澳洲拉筹伯大学的兼職教授。他的主要研究領域包括拓撲群、拓撲學和超越數論，同時對早期肌肉萎縮疾病的早期檢測、健康信息學和預測澳洲股票交易等方面也有廣泛的興趣。他是幾本書的作者。

Arthur Jones [1934-2006]和Kenneth R. Pearson [1943-2015]是澳洲拉筹伯大学的數學教授。他們對教學和數學都有極大的熱情。