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Abstract Algebra

Allen Hibbard
Organization: Central College
Department: Mathematics and Computer Science
Education level


Main text: Contemporary Abstract Algebra by Joe Gallian

Laboratory supplement: Exploring Abstract Algebra with Mathematica by Ken Levasseur and Allen Hibbard. This Mathematica-based book does not make any assumptions about the main text being used. It can readily be used by those who prefer a rings-first approach to algebra (eg., Hungerford users). Exploring Abstract Algebra with Mathematica includes 27 notebooks that are meant as labs for group and ring theory. In addition, it contains the documentation for the packages (AbstractAlgebra) upon which the labs are based.

This two semester Abstract Algebra course is an introduction to abstract mathematical systems, including groups, rings, and fields (and morphisms between these). This is a junior/senior level course intended for majors. It is assumed that students have already been introduced to various methods of proof and they are expected to write many proofs in this course. Partially due to its abstractness and demand for providing proofs, this is typically one of the more difficult courses for our major.

I am currently in the middle of the second time I have taught this year-long course with the Exploring Abstract Algebra with Mathematica notebooks. Based on anecdotal evidence, there are two main points noted by students. First, most students have appreciated the visualizations incorporated in the labs that help them to better understand the theoretic concepts. The graphics help not only those who are visual learners, but they also help encourage those who think symbolically to also learn visually. Second, after students have learned the computational basics, they have appreciated that the packages provide good computational tools for doing more complicated calculations.

While our students have had some previous exposure to using Mathematica, there is no need to have any programming experience to utilize the labs. The notebooks are used to either introduce a new mathematical concept or to reinforce concepts previously seen in class. Since the labs often motivate and introduce a new concept, the amount of lecture time is reduced accordingly. Students are encouraged to use Mathematica for any questions involving computations, as long as they understand how to do it on paper equally well.

*Mathematics > Algebra > Field and Ring Theory
*Mathematics > Algebra > Group Theory