Studying abstract algebra can be an adventure of awe-inspiring
discovery. The subject need not be watered down nor should it be
presented as if all students will become mathematics instructors. This
is a beautiful, profound, and useful field which is part of the shared
language of many areas both within and outside of mathematics.
To begin this journey of discovery, some experience with mathematical
reasoning is beneficial. This text takes a fairly rigorous approach to
its subject, and expects the reader to understand and create proofs as
well as examples throughout.
The book follows a single arc, starting from humble beginnings with
arithmetic and high-school algebra, gradually introducing abstract
structures and concepts, and culminating with Niels Henrik Abel and
Evariste Galois' achievement in understanding how we can-and
cannot-represent the roots of polynomials.
The mathematically experienced reader may recognize a bias toward
commutative algebra and fondness for number theory.
The presentation includes the following features:
- Exercises are designed to support and extend the material in the
chapter, as well as prepare for the succeeding chapters.
- The text can be used for a one, two, or three-term course.
- Each new topic is motivated with a question.
- A collection of projects appears in Chapter 23.
Abstract algebra is indeed a deep subject; it can transform not only the
way one thinks about mathematics, but the way that one thinks-period.
This book is offered as a manual to a new way of thinking. The author's
aim is to instill the desire to understand the material, to encourage
more discovery, and to develop an appreciation of the subject for its
own sake.
