*Introduction to Applied Algebraic Systems* is book of breadth that can support two semesters of lecture and study. This could be at the upper-level undergraduate or first-year graduate level. The title may suggest to some that applications are germane to the content. However, after a quick overview of UPC codes to exemplify applied modular arithmetic in Chapter 1, tightly coupled applications do not return until the final seventh chapter, on elliptic curves.

The core five chapters are a rigorous introduction to number theory, encompassing rings, fields, polynomial theory, groups, and algebraic geometry. This all leads to and supports two chapters on elliptic curves. Thus the book delivers a number theory foundation from modular arithmetic through basic fields on to algebraically closed fields, affine varieties and ideals. Each section concludes with about a dozen exercises well-motivated by the previous proofs and comments. The independent reader will find no solutions for these, but the frequent introduction of exercises after new ideas makes this manageable.

The final chapter, the second one on elliptic curves, is entitled “Further Topics Related to Elliptic Curves.” The reader that has made it this far will be prepared to understand and receive a glimpse of the application of these curves to cryptography, their use in prime factoring, and the role they played in Wiles’ proof of Fermat’s Last Theorem. The final topic introduces the idea of the genus of a curve.

Future computer scientists and number theorists interested in applications will find this book a good foundation text. Also, this is an excellent source for the advanced undergraduate seeking to make the transition to higher mathematics.

Tom Schulte first met elliptic curves in graduate study at Oakland University. He lives, works, and teaches in Michigan.