Authors:
- Motivates the development of algebraic concepts through tantalizing geometric questions from history
- Illustrates the power of algebraic abstraction for tackling concrete questions
- Engages the reader with abundant examples, commentary, and exercises
Part of the book series: Undergraduate Texts in Mathematics (UTM)
Part of the book sub series: Readings in Mathematics (READINMATH)
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Table of contents (12 chapters)
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Front Matter
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Back Matter
About this book
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.
Authors and Affiliations
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School of Mathematical and Physical Sciences, La Trobe University, Bundoora, Australia
Sidney A. Morris
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(Deceased), Victoria, Australia
Arthur Jones, Kenneth R. Pearson
About the authors
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.
Bibliographic Information
Book Title: Abstract Algebra and Famous Impossibilities
Book Subtitle: Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations
Authors: Sidney A. Morris, Arthur Jones, Kenneth R. Pearson
Series Title: Undergraduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-031-05698-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-031-05697-0Published: 28 November 2022
Softcover ISBN: 978-3-031-05700-7Published: 28 November 2023
eBook ISBN: 978-3-031-05698-7Published: 26 November 2022
Series ISSN: 0172-6056
Series E-ISSN: 2197-5604
Edition Number: 2
Number of Pages: XXII, 218
Number of Illustrations: 29 b/w illustrations
Topics: Field Theory and Polynomials, History of Mathematical Sciences