Customer Reviews

Abstract Algebra and Solution by Radicals

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This book introduces group theory and all the math needed to prove one of the central results of Galois theory, the insolubility of the quintic. This includes prioving many ruler&compass constructions in geometry are impossible.

That sounds heavy but the remarkable thing is anyone who has taken grade 12 math should be able to follow it (with a bit of work) and anyone who has done first year algebra or calculus should be able to follow it all.

Very discursive, with a lot of sentences not just symbols to explain the ideas, and a lot of examples. Nice physical layout too.

A hard core math text written for non-mathematicians, and it succeeds. I also highly recommend it to anyone encountering groups or Galois theory for the first time.

I taught out of the hardcover version of this book at SUNY College at Oneonta many moons ago. It was a course for first-semester sophomore mathematics majors. The goal of the book is to develop the subject matter that is needed to prove that the fifth degree polynomial is not solvable by radicals, i.e. there is no analogy to the Quadratic Formula for the quintic. (There is for the cubic and the quartic.) We had a good time because the course was focused on this one goal and the class knew exacly where we were headed. We got there, too. It was very easy to teach from this book and students rated it very highly. The students that I taught had Calculus I and II and a Foundations course as freshmen. The prior exposure to logic, sets and methods of proof (development of the integers) was very helpful, as was the maturity gained from the calculus (although the subject matter of Calculus was not necessary). I supplemented by rigorously developing the rational numbers, saving the reals for the Intro to Analysis course that followed this one. I am now a biostatistician outside of academia, but I hope that professors who are now teaching Abstract (Modern) Algebra will consider using this text in paperback form.

This charming little introit to abstract algebra is keyed on a theme of the algebraic equation, and the discovery of the insolubility of the quintic. This includes the history and final plight of the circle-squarers, and some of the history of Galois and Abel, working heroically and heuristically in the early nineteenth century without the recent easier access to the subject now available.
All math is divided into three parts, analysis, algebra, and topology and abstract algebra is no doubt abstract, but less so than analysis, and shows the beautiful hidden sructure behind number systems, from monkey-see monkey-do to counting on your fingers, to the square root of minus one and beyond. The progression from simple groups, to rings, and fields and the rest is a revelation of the complexity behind simple things and it is a pity the educational system cannot bring more to these vistas, where the elegant Galois theory caps the summits. A good book to amateurize with, and with a good mouse-hole entry for a look-see to the ultra-clever Galois theory. Superb.

This book provides the shortest journey from group theory to Galois theory (which determines when a polynomial equation can always be solved by means of an algebraic formula). Any topic in abstract algebra that isn't absolutely essential to this "trip" has been omitted from this book. Hence there's no coverage of ideals of rings, Euclidean domains, etc. The coverage of Galois theory is very rushed and proves only half of the theorem that is the book's ultimate goal; namely, that a polynomial equation can be solved by means of an algebraic formula if and only if the equation's Galois group is solvable. Some of the important proofs later in the book are sketchy.

The reader should be very familiar with methods of proofs and should have had some previous exposure to abstract algebra, especially groups and vector fields. The treatment of Galois theory is so rushed that, for self-study, the reader should refer to a second book on abstract algebra that treats Galois theory -- such as Richard Dean's Elements of Abstract Algebra (1966), which is the basis of the Maxfields' book, or W.E. Deskins' Abstract Algebra (1996).

This book in an excellent choice. Well written not complicated lingo, and for visual learners the diagrams are intelligently selected. Very good introduction to group theory, with many examples. I am a computer programmer so I did not want a book with pure math and this book uses everyday examples to make a point.

I am struggling with this book. There are times when more explanation is needed and yet, at other times, there is too much.

There is a lot of good information here and a lot of good exercises. And at a great price, I highly recommend this little book.