Abstract algebra is an essential part of the undergraduate mathematics curriculum, but students often focus on the details of proofs and miss the connections made between abstract algebra and other parts of their mathematics studies. For many this may be because of the anxiety that comes from dealing with abstraction for the first time. Add to this the ongoing debate of whether the course should start by dealing with groups or rings: it might be tough to decide which textbook to use for this very important course.
Introduction to Abstract Algebra by Fine, Gaglione, and Rosenberger attempts to be that one book that deals with all of these issues. This text contains the standard material for an undergraduate course in abstract algebra, with typical exercises. While it is intended to be used for a course that deals with rings first, it is flexible enough to use in a course that covers groups first. It contains more material than could be covered in a semester, and would be a nice choice for a year long course on abstract algebra.
At first glance, it seems a standard introduction to abstract algebra. It contains the usual content with a fair number of typical exercises for each section. Most of the examples are condensed into sections as opposed to being interspersed throughout the presentation of definitions and theorems. Upon closer inspection, however, we find that the organization of the text has some slight variations from other comparable books. For example, the first two chapters of this book deal with algebraic reasoning and algebraic preliminaries. From the beginning, the authors deal with the algebraic method. They frame the process around four main questions and four secondary questions. This seems to be a nice way for students to understand how to approach the course.
Throughout the text, presentation of the content connects the essential ideas of abstract algebra with ideas found in courses that most students would encounter earlier in their education. For example, content normally found under section titles such as “The Integers” or “Arithmetic in Z” is gathered into a chapter on number theory. In the chapter on fields, the authors include the constructions of the rationals and the reals. They do so in a way that connects ideas with which students are familiar to the process of abstracting these ideas. They include a section in which they review ideas of linear algebra so that they can be used to consider Galois theory.
This text book is a wonderful introduction to abstract algebra. The relationships between the ideas of the course and ideas found earlier in the undergraduate mathematics curriculum are reinforced throughout the text. This would make it a nice option for an independent study of abstract algebra. It is a nice reference book for a student who intends to continue the study of mathematics at the graduate level. I would recommend this book to those who teach undergraduate mathematics, whether for use in teaching the course or simply as a reference that can be used throughout the undergraduate curriculum.
Suzanne Caulk is an Associate Professor of Mathematics at Regis University in Denver, CO. She is very interested in modular forms and mathematics education.