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Abstract algebra with applications
Terras A., Cambridge University Press, New York, NY, 2019. 328 pp. Type: Book (978-1-107164-07-9)
Date Reviewed: Sep 18 2020

Any experienced teacher of a classical field of mathematics who wishes to publish a new textbook in such an area has to figure out how to present a topic with a long tradition to a new generation of readers. One approach is to present examples that show how the topic of interest is used in practice; Audrey Terras follows this idea in her book. Thus, teachers of abstract algebra can use this work as bait to convince students to study the field, even if they are not planning to pursue a mathematical career.

The book is divided into two parts. Part 1, “Groups,” introduces various aspects of groups (starting from the beginning) and includes some related applications. Part 2, “Rings,” deals with these types of structures (with some applications), but also extends its presentation to fields and vector spaces.

Part 1 is composed of four chapters, beginning with where algebra can be applied (for example, for describing various regular structures) and then elaborating on basic notions (various types of numbers, sets, mathematical methods of reasoning, and operations, including modulo arithmetic). Then, in chapter 2, the definition of a group along with various examples are discussed. Readers with an information and communications technology (ICT) background, where a lot of emphasis is put on numbers, are presented with some issues that they may not be used to (such as symmetries of geometric figures). Obviously, residue classes (along with all of their properties) are elaborated as well. Next, chapter 3 goes deeper with a presentation of various groups (permutation groups, isomorphisms, cosets, quotient groups, products of groups). The final chapter of this part deals with some practical examples of using abstract algebra: ciphers, finite Fourier transforms, conservation laws, and puzzles. They are either shown to be able to be described with the group apparatus or are enabled by some algebraic results (as in the case of the RSA cryptosystem).

Part 2, also consisting of four chapters, deals with problems where the introduction of a second operation (that is, potential to define a ring) is necessary. This way, chapter 5 shows the basic intuitions and gives fundamental definitions (except for rings and subrings, integral domains, fields, and polynomial rings; additionally, some notions are extended, for example, instance quotient rings). The subsequent chapter elaborates on various additional concepts: ring homomorphism, the Chinese remainder theorem, and polynomial rings. Chapter 7 extends the book’s presentation on vector spaces, matrix theory, linear mappings, and finite fields (with their extensions) and Galois theory. The last chapter again provides a bunch of examples, this time mostly related to the ICT field: generation of random numbers, linear error-correction codes, quantification on vertex importance in graphs (one of the aspects of the famous Google ranking for web pages), and cryptographic methods based on elliptic curves.

The book is written clearly with many examples. Almost all important notions presented are appended with exercises for self-study. I’d say that this book is aimed at readers who would like to gain a general knowledge of algebra (for example, first-year students) and who are also somehow interested in the general applications of this field. The book could also interest engineering students, although there are some books that may be perceived as competitive [1,2]. Such alternative titles focus on ICT applications and are not that interested in a very strict presentation from a mathematical viewpoint. I do not want to juxtapose them to the reviewed book, since all of them have their merits: Terras’ book is more in-depth from a mathematical viewpoint, while [1,2] are more thorough and insightful when dealing with the practical usefulness of this mathematical field. Thus, I would recommend Abstract algebra with applications to readers with some initial interest who are looking to gain a thorough mathematical basis (since it is shown where the theory can be used and how exciting it might be); later they can look to books such as [1,2] for a richer presentation of engineering methods.

Reviewer:  Piotr Cholda Review #: CR147064 (2102-0031)
1) Moon, T. K.; Stirling, W. C. Mathematical methods and algorithms for signal processing. Prentice Hall, Upper Saddle River, NJ, 2000.
2) Stepanov, A.; Rose, D. From mathematics to generic programming. Pearson, Upper Saddle River, NJ, 2015.
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