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＜E-Book＞
Abstract Algebra and Famous Impossibilities : Squaring the Circle, Doubling the Cube, Trisecting an Angle, and Solving Quintic Equations / by Sidney A. Morris, Arthur Jones, Kenneth R. Pearson
(Readings in Mathematics. ISSN:29455847)

Edition	2nd ed. 2022.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2022
Language	English
Size	XXII, 218 p. 29 illus : online resource
Authors	*Morris, Sidney A author Jones, Arthur author Pearson, Kenneth R author SpringerLink (Online service)
Subjects	LCSH:Algebraic fields LCSH:Polynomials LCSH:Mathematics LCSH:History FREE:Field Theory and Polynomials FREE:History of Mathematical Sciences
Notes	1. Algebraic Preliminaries -- 2. Algebraic Numbers and Their Polynomials -- 3. Extending Fields -- 4. Irreducible Polynomials -- 5. Straightedge and Compass Constructions -- 6. Proofs of the Geometric Impossibilities -- 7. Zeros of Polynomials of Degrees 2, 3, and 4 -- 8. Quintic Equations 1: Symmetric Polynomials -- 9. Quintic Equations II: The Abel–Ruffini Theorem -- 10. Transcendence of e and π -- 11. An Algebraic Postscript -- 12. Other Impossibilities: Regular Polygons and Integration in Finite Terms -- References -- Index 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 HTTP:URL=https://doi.org/10.1007/978-3-031-05698-7

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