When reviewing a text, the question “How well will the students understand the subject just by reading this book?” always runs through the back of my mind. And according to this criterion, Contemporary Abstract Algebra by Joseph Gallian is a well-written book. After introducing a new concept or theorem, he always provides a plethora of examples. He explains, “The best way to grasp the meat of a theorem is to see what it says in specific cases.” Furthermore, Gallian keeps the big picture in mind. Rather than just proving theorem after theorem, he often steps back to explain why a theorem is important or how the theorem can be used, or even to explain what a theorem means in words, rather than just in symbols. Another plus are the numerous tables he includes, which sum up a lot of information in a concise way. For example, there is a summary of group examples and their properties.
What makes Contemporary Abstract Algebra unique is Gallian´s focus on showing that abstract algebra is a contemporary subject. He incorporates examples of physics, cryptography, chemistry, and computer science into the text. For instance, there is a description of how your credit card number is encrypted when buying online from Amazon.com. In another section, he explains how molecules with chemical formulas of the form AB4, such as methane (CH4), have the same symmetry as the group A4. Gallian also shows that abstract algebra is an ever-expanding field of research by telling stories of how recent mathematicians pushed to solve certain problems. For example, he gives the history of “the enormous effort put forth by hundreds of mathematicians” since the 1960s “to discover and classify all finite simple groups.”
Contemporary Abstract Algebra is appropriate for a first or second course in abstract algebra. The text does not spend much time on preliminary number theory topics, like the division algorithm or modular arithmetic, so the students need to have familiarity with these topics. The text does provide a solid introduction to the traditional topics of groups, rings, and fields, but there is depth in his coverage of these topics. He includes, for example, internal and external direct products and the Fundamental Theorem of Finite Abelian Groups, in his section on groups. The special topics cover a selection of interesting themes, such as Cayley digraphs, which “provide a method of visualizing groups” and cyclotomic extensions which tie together many themes explored in the text.
The text also has some interesting extras. Gallian starts and ends each chapter with quotes from famous mathematicians, popular songs, and even few from Homer Simpson. Some of the quotes are simply amusing (“If you really want something in this life, you have to work for it — Now quiet, they’re about to announce the lottery numbers.” –Homer Simpson). Others offer sound mathematical advice (“‘For example’ is not proof.” –Jewish proverb). At the end of each chapter, he offers a list of suggested readings with summaries, as well as suggested websites, and films. The text also has a companion website which has true/false questions and software for computer exercises.
In the preface, Gallian lays out his goals for the text. Briefly, they are to give the student “a solid introduction to the traditional topics,” to show readers that “abstract algebra is a contemporary subject,” to provide students with an enjoyable text, and finally to help students gain competency in doing computations and writing proofs. He definitely meets all of these goals, and as such, I certainly recommend this textbook.
Kara Shane Colley studied physics at Dartmouth College and math education at Teachers College. She is currently taking a break from teaching math to volunteer at a meditation center in Mexico.