This old standard (the first edition came out 25 years ago) has been nicely refreshed. I have taught successfully (or so I hope) out of several editions of this text and have yet to find a better choice for a first undergraduate course in abstract algebra. It is written in a style which is both rigorous and welcoming. Anyone who knows Gallian will find his personality on almost every page.

The degree of difficulty and topic order of this text seems ideal for the majority of students. Careful choice of exercises and chapters on the part of the instructor can vary this difficulty level enough to make this text a safe choice for all but high-level honors sections. For example, Chapter 0 provides a nice introduction to the integers and equivalence relations. If your department offers a bridge course, you might skip this chapter.

Gallian treats several topics which most texts omit or treat in much less detail. Among these are Geometric Constructions, Finite Simple Groups, Frieze Groups and Crystallographic Groups, and Algebraic Coding Theory. While most of us won’t be able to cover all of these topics in to a two-semester course, they provide lots of options for an honors section or as starters for student honors projects.

According to its introduction, the new edition contains 200 new exercises, new examples and an updating of the quotations (which are wonderful), historical notes and biographies. Those features are among the many strengths of Gallian’s text and the updates are well done.

The answers to selected exercises are more than just answers providing what Gallian refers to as “skeleton solutions and hints.” Additionally the text now comes with most of the ancillaries we have come to expect from Calculus texts including:

Student Solutions Manual. Instructor’s Solution Manual. Online Solution Builder which allows instructors to create solutions printouts in PDF. Website with student resources including T/F questions with comments, flashcards, essays on learning abstract algebra.

The exercises range from simple computations to very interesting theoretical problems, many of which provide previews of theorems to come. The number of exercises on cyclic groups has gone from 69 to 86. Several of the added problems were simple computations of the sort I would assign when I used earlier editions. Here are two examples:

57) Determine the orders of the elements of D_{33} and how many there are of each.

72) Let a be a group element with |a| = 48. Find a divisor k of 48 such that

a) (langle a^{21} angle = langle a^{k} angle )

b) (langle a^{14} angle = langle a^{k} angle )

c) (langle a^{18} angle = langle a^{k} angle )

Here are a couple of more theoretical exercises that were added. From Chapter 14:

43) If (R) and (S) are principal ideal domains prove that (Roplus S) is also a principal ideal domain.

From Chapter 18:

44) Let F be a field and R be the integral domain in F[x] generated by x^{2} and x^{3} (that is, R is contained in every integral domain that contains x^{2} and x^{3}). Show that F is not a unique factorization domain.

Several of the photos/images of mathematicians that appear at the end of most chapters have been updated. I quickly scanned back to the chapter on finite simple groups to see what had happened to the photos of Aschbacher, Gorenstein, and Thompson. Thompson’s photo had not changed, but Aschbacher’s is now much more current. Interestingly, Thompson’s appears to be a much older shot of him as a young man working with a student. It would be nice if the photos and quotes were dated, but that is a very small quibble.

Abstract Algebra 8^{th} Edition is a very solid text which is suitable for a wide range of students. In addition, it now has all the bells and whistles we expect in a main-line calculus text. It is highly recommended. Indeed, even if you don’t choose to use it for your course, you should have a copy on hand to peruse for quotes, good examples and great mini-biographies.

Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences at North Central College in Naperville, IL.

## Comments

## Note the price

I think that one thing that needs to be mentioned about Gallian's book is the absolutely ridiculous price. The price quoted at the beginning of this review is $236.95, which I think is unconscionable.

Mark Hunacek## Price of book

The current publisher's list price (February 2015) is $185.95. eBook price is $55.49. The Amazon price is $129.00. Cengage rental price is $33.00.

## Price listing changed!

Price listing changed!