# Norman R. Reilly: Introduction to Applied Algebraic Systems.

NORMAN R. REILLY: Introduction to Applied Algebraic Systems by, Ph.D. Oxford University Press, 2010. ISBN: 978-0-19-536787-4.

The main objective of the author in this text is to introduce many algebraic concepts (groups, rings, etc) that are typical in a first and/or second course in abstract algebra, while motivating the concepts with various real-life current-day applications. The author believes that the presentation flows more naturally if rings are introduced before the topic of groups by way of number theory. The topics are discussed in a rigorous manner with a colloquial style that the reviewer found very enjoyable.

The algebra topics covered in the book are: modular arithmetic, rings and fields, groups and permutations, group homomorphisms and subgroups, polynomial rings and algebraic geometry (mainly elliptic curves). Each topic flows naturally from the preceding topic, with most chapters finishing with an application of the ideas recently presented. With attention paid to storing, securing, retrieving and communicating electronic information, the applications discussed are: bar codes, public-key cryptosystems, error-correcting codes, colorings, enumeration, the Enigma encryption machine and elliptic-curve cryptosystems.

The reviewer was particularly impressed with the flow and depth provided for each application, and found the discussions fascinating. For almost all the applications, the author presented the "big-picture idea" of the application first together with a need for the application, and then switched to discussing its details; this style promises to be very helpful to students. In some cases, a significant amount of history was provided to help place the application in context. The discussion on Fermat's Last Theorem was very well presented, including a mathematical overview of Wiles' proof, with references provided if more detail is desired.

The notion of a ring is introduced by describing it as a natural generalization of the concepts discussed in the previous chapter on modular arithmetic. This is skillfully and seamlessly achieved by the author, so that this deviation from standard algebra texts loses nothing by doing so.

The book is suitable for any student with a sound knowledge of linear algebra. Those students who additionally have some maturity with proofs should be very comfortable with the text. The reviewer found only three very minor typographical nonmathematical errors while reading the book, so students should be pleased with the typesetting. For instructors using the book in a class, there is a solutions manual available from the author for instructors. The author provides many resources for extra reading by providing references where applicable.

In summary, the book is ideal for a thirty-week (that is, two-semester) course in abstract algebra, providing students with a sound background in algebra, while simultaneously teaching them many current-day uses of algebra in the real-world.

Michaela Vancliff, Math Dept, Univ of Texas at Arlington