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Problems in Abstract Algebra

A. R. Wadsworth
Publisher: 
American Mathematical Society
Publication Date: 
2017
Number of Pages: 
277
Format: 
Paperback
Series: 
Student Mathematical Library 82
Price: 
52.00
ISBN: 
9781470435837
Category: 
Problem Book
[Reviewed by
Fernando Q. Gouvêa
, on
05/17/2017
]

It is a truth universally acknowledged that no teacher is completely happy with the textbook. A common complaint is that the problems are not the right ones: they are either too easy or too hard, they don’t have the correct focus, they don’t include our favorites. So many of us end up supplementing the problems found in the book with other problems, either of our own creation or found (where else?) in other books. This collection of problems in Abstract Algebra came about exactly that way.

The author tells us that the problems in the textbook or textbooks he was using were not interesting enough and not very demanding, so he supplemented them. (I do the same. Maybe we all do.) These supplementary problems are the origin of this book. The book also includes short summaries of background material, mostly to establish notation and nomenclature (e.g., what is the “First Isomorphism Theorem”?).

Problems in algebra can be of many kinds. Some demanding problems involve specific examples and difficult computations, while others are “little theorems,” results that could be included in a textbook but often are not. This book tends toward the latter. Even some “big theorems” are included (e.g., the Nullstellensatz is problem 5.37).

For a sampler, here is a summary of the first four problems in the section on “Rings, subrings, and ideals”

  1. Adding a 1 to a “non-unital ring.” (For some reason, this is numbered 3.2, but I cannot find a 3.1)
  2. Finding the two-sided ideals of the matrix ring \(M_n(R)\). (I would have asked for a left ideal as well.)
  3. If an element \(a\in R\) has a right inverse but not a left inverse, then it has infinitely many left inverses.
  4. If \(a^3=a\) for every \(a\in R\) then \(R\) is commutative.

The first three are clearly of the “little theorem” species. The fourth initially seems to call for some sort of trick, but it is an instructive trick. Similarly, the section on Galois theory includes Artin’s proof of the Fundamental Theorem of Algebra and the theorem on symmetric functions, both real theorems. But there is also a monster like

Let \[K=\mathbb{Q}(\sqrt{(5+\sqrt5)(21+\sqrt{21})}).\] Prove that \([K:\mathbb{Q}]=8\) and that \(K\) is Galois over \(\mathbb{Q}\) with Galois group isomorphic to the quaternion group of order \(8\).

That one too involves an “instructive trick.”

The topics covered in the first three chapters are groups and rings (which includes a little bit on fields), basically the material in a one-semester abstract algebra course. The fourth chapter does “advanced linear algebra”: dual spaces, quotient vector spaces, canonical forms, and a little bit of matrix groups. The fifth chapter covers more advanced field theory and Galois theory. The author says this reflects the one-year algebra course taught at UCSD.

There are no solutions.

The guiding philosophy for this is that readers who do not succeed with a first effort at a difficult problem can often progress and learn more by going back to it at a later time. Solutions in the back of the book offer too much temptation to give up working on a problem too soon.

That is most certainly true, but for some instructors pressed for time it will limit the usefulness of the book.

Pet peeve: the author uses \(\mathbb{Z}_d\) to denote \(\mathbb{Z}/d\mathbb{Z}\). This is an evil practice, since that notation should denote the \(d\)-adic integers. (Yes, I know this is a lost cause, but I can’t help it.)

Nitpick: the cover is very boring.

I’ll certainly be using some of the problems the next time I teach algebra. I’m even tempted to make it the only textbook for the course.

 

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Fernando Q. Gouvêa has taught introductory algebra many times, but has never been happy with the textbook he chose.

See the table of contents in the publisher's webpage.