Modern Algebra with Applications

It has been some time since I taught a course in modern algebra, but if I were to teach it again, I would use this book. The first chapter covers the basics of binary operations and algebraic structures. Seven pages in length, it is a basic overview of what the students should already know. Chapter two covers what is arguably the simplest abstract algebra, Boolean algebra. Propositional logic, switching circuits, posets and lattices are the examples used. I am in strong agreement with the introduction of switching circuits this early in a course. It is an application of abstract algebra that students can easily understand and relate to. Switching circuits also immediately demonstrates that the inclusion of "with applications" in the title is not a misnomer. Basic group theory is the topic of chapter three, quotient and symmetry groups are covered in chapters four and five. The Polya-Burnside method of enumeration is the topic of chapter five, monoids and machines the focus of chapter seven, rings, fields, polynomial, Euclidean and quotient rings are examined in chapters eight, nine and ten. Field extensions are covered in chapter eleven, Latin squares in chapter twelve, geometrical constructions in chapter thirteen and error-correcting codes in chapter fourteen. The exposition is excellent, it is one of the more readable texts in advanced mathematics that you will find. There are many examples of the applications of modern algebra to the real world and a large number of exercises are included at the end of the chapters. Solutions to the odd numbered exercises are found in an appendix. Modern algebra is a difficult topic due to the abstract nature of the material. It is easy for students to get confused and lost when they are studying the book on their own. The risk of that is minimized if they are reading this book.

This book is pitched at the level of a third or fourth undergraduate year in maths, or at a graduate level course. Gilbert assumes a solid background in maths, especially linear algebra. He expounds on groups and rings. You can certainly treat this book as a pure maths text on algebra. But there is a twist to it, for some of you. He discusses applications in engineering. Notably Boolean algebra. Boolean logic forms the conceptual and practical basis of computer hardware and software. Gilbert shows how to represent a Boolean expression in a normal form and how to simplify it. Very practical and vital to circuit design. Later on in the book, he gives a good but all too brief discussion of finite state machines. These can be used in control systems and even in the construction of computer languages like Java. Where the Java bytecode can be validated precisely because Java is a finite state machine. Finally, error correcting codes take up an entire chapter. Important in communications theory, and even in such common items as storing information on computer media like DVDs.