Selected Exercises in Algebra, Volume 1

Selected Exercises in Algebra is a planned two volume series. This review is for volume1. As the title suggests, the book contains exam exercises in algebra for the Arithmetic courses taught at the University of Pisa. Included are notes detailing the theory to be used along with the worked out solutions. Each problem is unique and two solutions accompany each problem. There are eight major themes from which the exercises are derived from: Mathematical Induction, Combinatorics, Modular Arithmetic, Abelain Groups, Communtative Rings, Polynomials, Field Extensions, and Finite Fields. The book contains a section of relevant theory, which provides a detailed reference for study and a list of preliminary exercises that provide an introduction to the main topics and solution techniques to be used for solving the problems. Lastly, there is a list of problems recommended for exams.

 

We, as mathematicians, know the value of practicing solving problems, proving theorems, and drill and practice. It is the one, and only, true way to learn mathematics.  Only by grinding, staying the course, and having a diverse selection of problems from different fields of mathematics, do we gain the “intelligent eye” for how to solve a variety of problems.  The main reason why the authors wrote this book is, “the idea that Mathematics can only be learned by rediscovering it.” It is through this process of using critical thinking and logical skills; we develop creative ideas and make connections to successfully get answers.

 

There are also other problems that test students' ability to creatively use algebra in a proof. One of these examples is below:

 

 

The authors present four different solutions – first by solving the problem by means of some properties of binomial coefficients, by mathematical induction, using a combinatorial solution, and using the binomial theorem.  It is wonderful to see the diversity of various ways to show the proof. This will expose students to different techniques and brings in diverse ways to approach problems.

 

This book is filled with rich exercises with detailed historical notes mirroring Arithmetic and Algebra. It gives the students a way to practice the art of proofs and logical thinking. I can see this book being used in many ways ranging from class projects, practicing additional problems to supplement another text, or helping students get ready for an exam. This book can be used for both the professor who needs a reservoir of problems for homework or projects and for the student who wishes to tackle more exercises for drill and practice. The book left me thinking that the more you practice and know, the better.

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Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His Research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at pto2@psu.edu.