This is an application for those who want to learn or teach college level abstract algebra. It was constructed at a small Midwestern college in order to incorporate the latest information on design of instruction (DI), and to avoid the unnecessary costs associated with proprietary applications. The software is open source, free of charge, and can be downloaded here: www.omsaa.org . While it has only been used in one college setting (and thus is not yet proven on a broader basis), our experiences have been very positive.

Abstract algebra is the branch of mathematics that studies algebraic structures. These are often defined by sets of axioms relating the operations defined on these sets. This book, which was written as part of a senior-level undergraduate course at Wellesley College, provides some examples of these axiomatic systems and their associated structures—groups, rings, and fields—and shows how to prove basic results about them. Along the way we give many example problems for students to work on and illustrate their solutions with computer programs written in JavaScript. The text is interspersed with hints for getting the computer programs to work.

## Ancillary Material

### About the Book

This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.

Until recently most abstract algebra texts included few if any applications. However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation.

This text contains more material than can possibly be covered in a single semester. Certainly there is adequate material for a two-semester course, and perhaps more; however, for a one-semester course it would be quite easy to omit selected chapters and still have a useful text. The order of presentation of topics is standard: groups, then rings, and finally fields. Emphasis can be placed either on theory or on applications. A typical one-semester course might cover groups and rings while briefly touching on field theory, using Chapters 1 through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the first part), 20, and 21. Parts of these chapters could be deleted and applications substituted according to the interests of the students and the instructor. A two-semester course emphasizing theory might cover Chapters 1 through 6, 9, 10, 11, 13 through 18, 20, 21, 22 (the first part), and 23. On the other hand, if applications are to be emphasized, the course might cover Chapters 1 through 14, and 16 through 22. In an applied course, some of the more theoretical results could be assumed or omitted. A chapter dependency chart appears below. (A broken line indicates a partial dependency.)

The 2021 Annual Edition is now available. Electronic editions have been updated. Print is being made available at online retailers – see the Purchase page for the latest details.

Sage is an open-source program for doing mathematics and is the ideal companion to Abstract Algebra: Theory and Applications. Sage is designed to be a free, open source alternative to Magma, Maple, Mathematica and Matlab. It includes many mature and powerful open-source tools for mathematics, such as GAP for group theory. With a strength in number theory, Sage also has excellent support for rings and fields.

Rob Beezer has contributed extensive material about studying abstract algebra concepts with Sage, and instruction in the use of Sage itself. Each chapter (except one, Matrix Groups and Symmetry

) has an extensive discussion of how to profitably use Sage. For most chapters (except two), these discussions are followed by classroom-tested exercises, ranging from very computational to open-ended guided explorations. In total there are 710 examples of Sage code and 121 exercises. These examples are run through automated testing twice a year using the latest stable version of Sage, so are highly reliable.

All of this material is included in the online version, where the Sage examples are executable and editable, via the free, zero-configuration Sage Cell server. The PDF download below is a static version of the online version and includes all the Sage material, where the examples have sample output included.

## Abstract Algebra

- An Inquiry-Based Approach to Abstract Algebra by Dana C. Ernst (Northern Arizona University). Free and open-source course materials for an abstract algebra course.
- Abstract Algebra: Theory and Applications by Tom Judson (Stephen F. Austin University).
- Elementary Abstract Algebra: Examples and Applications by Justin Hill (Temple College) and Chris Thron (Texas A\&M University-Central Texas) with contributions from others. The text is designed for students who are planning to become secondary-school teachers. The authors particularly emphasize material that has relevance to high school math, as well as practical applications. Much of the content is derived from the Judson book mentioned above.
- Essential Group Theory by Michael Batty (University of Durham).
- Group Theory: Birdtracks, Lie’s, and Exceptional Groups by Predrag Cvitanović (Georgia Tech).

## Linear Algebra

- A First Course in Linear Algebra by Rob Beezer (Puget Sound University). FCLA is a free and open-source introductory textbook designed for university sophomores and juniors. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The book is available for free in PDF and HTML form.
- Linear Algebra by Jim Hefferon (St. Michael’s College). This is another free and open-source textbook.
- Linear Algebra by David Clark (SUNY New Paltz). These notes are available for free from the Journal of Inquiry-Based Learning in Mathematics. The notes are designed for an inquiry-based learning (IBL) approach to the subject.
- Linear Algebra by Gilbert Strang (MIT). This isn’t really a textbook, but rather free course materials, including videos, available from the MIT Open Courseware Project.
- Paul’s Online Math Notes by Paul Dawkins (Lamar University).

## Conclusion

Abstract algebra is the study of sets and the primary object of study in abstract algebra is the set itself. What operations does it support? How many elements does it contain? How are the elements related to each other? This book presents the foundations of abstract algebra, as well as important concepts in number systems, groups, modules, rings, fields, etc.

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