Undergraduate Algebra: A Unified Approach

The book under review consists of two parts and the prerequisites. The prerequisites cover primarily some basic Set Theory and can be skipped by a reader already familiar with the notions; however, this section is important to keep the book self-contained. 

 

The two parts are very different by their content, style, and purpose, which make this book quite unique. The first part, The Language of Algebra, consists of chapters Glossary of Basic Algebraic Structures, Examples of Groups and Rings, Homomorphisms, and Quotient Structures.  The key novelty and unique feature of the book is that, in this first part, analogous topics on different algebraic structures are considered simultaneously (for example, the section on substructures introduces subgroups, subrings, subfields, etc., subsequently; or another example, the section on normal subgroups and quotient groups is followed by the section on ideals and quotient rings). This organization of the book is coherent and surprisingly efficient; it certainly provides a serious alternative to the traditional one where groups, rings, and fields are treated independently. It is interesting to compare the chapters by their styles; for instance, in Chapter 1, the reader can see only some simple examples that may already be familiar, but Chapter 2 consists completely of examples. Ultimately, the main purpose of Part 1 is to provide some basic language of algebraic structures in order to create necessary machinery for Part 2.

 

The second part, Algebra in Action, consists of the three chapters: Commutative Rings, Finite Groups, and Field Extensions. This is the more dynamic and mathematically appealing part of the book. The presentation is quite different from the first part, as the structure and exposition is closer to how mathematicians truly work. For example, the proof of the Fundamental Theorem of Finitely Generated Torsion Modules resembles the process in which technically involved results are split into smaller claims (in this particular case, seven claims). The book is written in a lively style and is pleasant to read. New concepts are carefully introduced, starting with an informal discussion and, occasionally, historical comments. Each section ends with exercises which progress from easy to challenging, and give a great deal of insight into the subject.

 

Some comparable books are Fraleigh's A First Course in Abstract Algebra, Herstein's Abstract Algebra, Hungerford's Abstract Algebra: An Introduction, and Rotman's A First Course in Abstract Algebra with Applications.  The first two books follow the standard ordering of groups, then rings, then fields.  On the other hand, Hungerford's book starts with rings first (the new editions provide an additional path starting with group theory), and Rotman's book starts with groups (but, as with Hungerford, also provides a path starting with ring theory). As mentioned above, Bresar's approach, which is based on a unified treatment of similar concepts across different algebraic structures, is very different from these four books. With a length of slightly more than 300 pages, this book is closer in length to the one written by Herstein and about the half the length of three other books.

 

I think it is greatly beneficial to the reader that, after the book starts with some familiar objects (integers, rational numbers, etc.) and formal definitions, the whole second chapter is spent dealing with examples. The author writes, ``the theory exists because of examples!'', and this is not just a slogan, but an attitude that structures the book in a way that helps with understanding abstract algebraic concepts and ideas. In my opinion, it is very convenient (especially for self-studying) that the exercises are ordered by their difficulty level.  Additionally, the last chapter (Field Extensions) is a bridge between undergraduate and graduate Algebra. From this perspective, this book can be also used by incoming graduate students to refresh their knowledge of Algebra before taking graduate courses.

 

I highly recommend this book for a standard undergraduate algebra course, as well as to students interested in independent study.

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Louisa Catalano is a graduate student at Kent State University. Her mathematical interests are Linear Algebra and Ring Theory and she is an active member of the Association for Women in Mathematics.