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Abstract Algebra and Famous Impossibilities

Sidney A. Morris , Arthur Jones , Kenneth R. Pearson
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Benjamin Linowitz
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Abstract Algebra and Famous Impossibilities aims to develop the abstract algebra necessary to prove the impossibility of four classical problems: trisecting the angle, doubling the cube, squaring the circle, and solving (in terms of radicals) the general quintic polynomial equation. The book is written in a very clear, friendly manner meant to make it accessible to students with limited previous exposure to proofs.
This book evolved from lecture notes for a course offered for many years at La Trobe University whose only prerequisite was a course in linear algebra covering, among other things, vector spaces over fields. It is therefore expected that the reader has had some exposure to the concept of a field, but nothing like what would be covered in a semester long course on Rings and Fields or Galois Theory. In light of this prerequisite, several introductory sections would likely be too terse for a reader without prior exposure to fields. For example, the authors discuss and define fields, finite fields, rings, and field extensions in less than four pages. Moreover, fields and rings are not defined formally, but only in vague terms (i.e., 'a set in which one perform the operations of addition, subtraction, multiplication, and division').
The first six chapters form the core of the text. Chapters one through four cover the field theory needed to discuss the irreducible polynomial of an algebraic number and finite field extensions. Chapter five contains a very well-written introduction to straightedge and compass constructions. This chapter concludes with a discussion of the field of all constructible numbers. In chapter six the authors give elegant proofs of the impossibility of trisecting the angle and doubling the cube. The proof that one cannot square a circle is of course more involved as it reduces to proving that \(\pi\) is transcendental. This is proven in chapter ten, along with the transcendence of \(e\). While building up to these proofs the authors discuss the solvability of quadratic, cubic, and quartic equations, and use Abel's original argument to prove the unsolvability of the general quintic equation.
I would be remiss not to mention Charles Hadlock's Field Theory and Its Classical Problems. Like Abstract Algebra and Famous Impossibilities, Hadlock's text introduces the abstract algebra necessary to prove the four classical impossibility problems. Both books are very well-written. (Field Theory and Its Classical Problems won the MAA's Beckenbach Book Prize in 1984.) Both books are roughly the same length. (Technically Field Theory and Its Classical Problems is 100 pages longer than Abstract Algebra and Famous Impossibilities, but the final 100 pages are devoted entirely to giving the solutions to all of the book's exercises.) Unlike Abstract Algebra and Famous Impossibilities, Hadlock's Field Theory and Its Classical Problems assumes no prior exposure to field theory or abstract algebra and begins by defining fields as subfields of the field of real numbers. I have found that this restriction makes Hadlock's introduction to field theory much easier to digest for students and prevents them from getting so distracted by field theoretic abstractions that they lose sight of the connections to the geometry of straightedge and compass constructions.
Abstract Algebra and Famous Impossibilities is a very interesting book. It covers the abstract algebra needed to give solutions to four of the classical impossibility problems, and in doing so connects algebra to some beautiful concepts from geometry. I think that it would make a wonderful supplementary text for a course on Galois Theory. That said, Field Theory and Its Classical Problems is one of my favorite mathematical texts (on any subject), and I won't be giving it up any time soon.
Benjamin Linowitz ( is an Associate Professor of Mathematics at Oberlin College.