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Essentials of Modern Algebra

Cheryl Chute Miller
Mercury Learning
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Talk to any college instructor who has taught a sophomore/junior level course in abstract algebra and ask what book he or she used; almost certainly the author’s name will be Gallian, Fraleigh, Rotman or Hungerford. Each of these names is associated with a well-known text that has been around for many years and gone through several editions. Though these books are all excellent and any one of them would be a good choice for an undergraduate course, it is worth noting that there are other, perhaps not as well known, choices available as well. Nicodemi’s An Introduction to Abstract Algebra with Notes to the Future Teacher and Reis’ Abstract Algebra: An Introduction to Groups, Rings and Fields have already been reviewed in this column, and the text by Miller that is now under review is also one that should be looked at by somebody teaching a one or two semester course in beginning abstract algebra.

This is a “groups first” book that has modest prerequisites; any student who is familiar with matrices and vector spaces should be fully prepared for this book. A glance at the table of contents reveals nothing terribly surprising; the selection of topics and order of presentation is fairly standard. Following an introductory “chapter 0” on basics like functions and equivalence relations, there are three chapters on groups (covering the basic examples, subgroups, cyclic groups, homomorphisms and isomorphisms, direct products, normal subgroups and quotient groups, and — very briefly and without proof — the theorems of Cauchy and Sylow and the fundamental theorem of finite abelian groups), followed by chapters on rings (the basic definitions, ideals and quotient rings, polynomial rings) and fields (extension fields, an introduction to Galois theory, and solvability). For those who care about such things, the author’s definition of “ring” omits the requirement of a multiplicative identity (a definition which is not uncommon at the undergraduate level, I think, but much less so in upper-level books on graduate algebra and ring theory) and the author draws a distinction between “integral domains” and just plain “domains” (the latter need not be commutative).

Though the topics presented are pretty standard, there are some interesting pedagogical ideas employed here, particularly the use of projects at the end of every chapter. The projects (each one consisting of a sequence of related, not particularly difficult, questions) vary in style: some are computational, some call for proofs, some require the student to identify errors in proofs, and some have the student fill in the blanks in a partial proof. For instructors who like to encourage group work, these projects would seem to be ideal (no pun intended).

I particularly liked the projects of the fill-in-the-blank variety, since they gave the student practice in filling in details without having to worry about producing an argument in the first place. I wish more undergraduate proof-oriented books employed similar kinds of exercises.

Some of these projects, it should be noted, consist of establishing results that many instructors would routinely present in class; for example, two projects investigate the groups of symmetries of an equilateral triangle and square, respectively. (Unfortunately the phrase “dihedral group” does not seem to appear in the text, even though there is an entire chapter devoted to certain important groups.)

Miller has obviously given a great deal of thought over the years as to how to present this material, and that time shows in the finished product. The exposition is succinct but clear, and it seemed to me that the author made a real effort to identify common student errors and address them in the text, such as pointing out, when certain theorems were proved, that their converse was not true. There are also, in addition to the projects, lots of homework exercises, all of them collected at the end of each chapter and grouped by section. These problems struck me as being on the quite easy end of the spectrum; many were calculational, and the ones that required proofs rarely if ever called for difficult ones. For this and other reasons (described shortly) this book may be particularly well-suited as a text for algebra courses at community colleges or other colleges where a more challenging text might be unsuitable.

In comparison to other books on this subject, this one seems slimmer and more compact, perhaps because the author has streamlined the text, omitting topics that often appear in competing texts but are rarely actually covered in a first course (or first two courses). So, for example, the reader will not find here any discussion of group actions, generators and relations, or modules. There is also no discussion of applications of abstract algebra such as coding theory, cryptography or general symmetry groups.

There were a few times, in fact, when I thought the author may have trimmed a bit too much, cutting out some muscle along with fat. Introductory number theory, for example, is seriously de-emphasized in this book. There is a brief summary of basic facts about divisibility (without proofs) in chapter 0, which consists of the standard facts about divisibility and prime numbers. I did not see the term “congruent modulo n” ever defined in the text, although the phrase “a (mod n)” is defined to be the remainder when the integer a is divided by n. This leads to the group \(\mathbb{Z}_n\) being defined not as a set of residue classes modulo n but as a set of actual integers (from 0 to n–1), with the sum of a and b defined to be (a+b) (mod n), which to my mind makes the proofs of associativity of addition and multiplication in \(\mathbb{Z}_n\) a bit more awkward than they perhaps need to be, and, more important, tends to obscure the reason why these operations should be associative. Some additional technical awkwardness occurs in the statement of Sylow’s theorem, where the phrase “r = 1 (mod p)” appears; obviously the author is attempting to convey the fact that r is congruent to 1 modulo p, but from her own definitions, “1 (mod p)” is actually just 1, so this statement really reads “r =1”.

When I last taught abstract algebra, I spent the first two full weeks of the semester talking about divisibility, primes and congruence modulo n, actually even proving the division algorithm (guaranteeing the existence of a quotient and remainder when one integer was divided by another non-zero one). This was time-consuming, but I thought the time was well spent, for several reasons: first, basic number theory was used throughout the semester and allowed what is, in my opinion, a more satisfactory definition of \(\mathbb{Z}_n\) as a set of residue classes. (Granted, this is a bit more abstract, but I found that if done slowly and carefully, it paid dividends down the road.) Doing some proofs in number theory also gave the student valuable practice in proving relatively routine theorems in a fairly natural setting, before the abstraction of the group concept kicked in, and, additionally, helped reinforce the idea that even routine and intuitively obvious theorems need to be proved. I would, therefore, have liked to have seen about ten pages or so of text added along these lines in the book under review.

I would also liked to have seen polynomials treated with a little more care. The author begins this chapter by stating that in it “we carefully introduce the notation for polynomials and roots of polynomials” but unfortunately doesn’t really give a definition of them, just treats them as formal sums of multiples of powers of some unknown thing x, about which the book says only “we must consider x to be a fixed symbol used only to define polynomials. We will not use it to stand for an arbitrary element of a set, or an unknown to solve for, from now on.”

It seems to me that a really good student can easily wind up feeling confused by a statement like this. Of course one can define a polynomial over a ring A rigorously to be an infinite sequence of elements of A, all but a finite number of which are 0; the mysterious x can then be defined to be the sequence (0,1,0,0,0,…). This is the way I learned the definition as an undergraduate, and it seemed perfectly sensible at the time. Admittedly, times have changed in the 40+ years since I first saw this definition, and perhaps there is some benefit in giving a definition of a polynomial as a formal sum, but I think that the student should at least be informed that things are not as simple as they might seem. In Peter Cameron’s undergraduate text Introduction to Algebra, for example, a polynomial is first introduced (in an introductory chapter) in the naïve formal way, but the student is explicitly warned that the question “what exactly is a polynomial?” is “much more difficult to answer than is indicated here”. The reader is then referred to the next chapter for a more detailed discussion, which begins with the statement “polynomials are easy to understand, but difficult to define; we must make the attempt.” I agree with this. Even if one disagrees with the statement that “we must make the attempt”, however, I believe we must at least inform the student that there are subtle issues that do need precise definition; of all the courses an undergraduate math major takes, I am hard-pressed to think of any (other than perhaps real analysis) in which precise definitions are so important.

There are, in addition, other omissions, which perhaps are not problematic for instructors teaching a one-semester course, but might be troublesome for people intending to use this book for a two-semester sequence. I have previously mentioned, for example, the omission of proofs of such “big” theorems as the fundamental theorem of finite abelian groups, and the theorems of Sylow and Cauchy. Not only are these theorems not proved, the author doesn’t really do much with them other than offer one or two examples. An instructor of a second-semester abstract algebra course might want to do more with these results.

An instructor of such a course might also want to cover principal ideal domains, Euclidean domains, and unique factorization domains; none of these are mentioned in the book. Presumably the author omitted these topics because she wanted to get to some basic field theory (Galois groups are defined and the Galois correspondence for finite fields and fields of characteristic 0 is established, and in a final chapter the notion of solvability is introduced and the impossibility of some classical constructions discussed); however, including more extended discussion of these omitted topics would have enhanced the flexibility of the book for instructors who think that, at this level, PIDs are more important than Galois groups.

(And speaking of Galois, writing that he “wrote down the basics of a theory that uses an interaction between groups and fields to help understand which polynomials can be solved ‘by radicals’” without at least mentioning the circumstances under which he wrote that down seems like a missed opportunity to tell the students something that many of them find really interesting.)

The modest length of this book does have some tangible advantages, though. As of this writing, it is selling on for about 65 dollars, which is less than half the cost of Gallian’s book (and in fact about ten dollars less than what charges just to rent Gallian’s book). More and more, as textbook prices soar upwards, I am starting to consider the price of a book in selecting a text to use for a course.

In summary, this is a competent, serviceable book, recommended particularly for courses at the lower end of the difficulty scale or courses where the students will do group work or papers or other activities that would allow maximum use of the projects.

Mark Hunacek ( teaches mathematics at Iowa State University.

Preliminaries. 1. Groups. 2. Special Groups. 3. Quotient Groups. 4. Rings. 5. Quotient Rings. 6. Domains. 7. Polynomial Rings. 8. Factorization of Polynomials. 9. Extension Fields. 10. Galois Theory. 11. Solvability. Bibliography. Index.