Math 71

Abstract Algebra
Last updated June 19, 2019 12:15:07 EDT

Course Objectives: If you tell your friends that you are taking an algebra course, they may patronizingly tell you they studied that in high school, and of course so did you. But still, there must be some connection, right?

In high school, algebra is about solving equations, often just linear equations. In linear algebra, you study properties of systems of linear equations and how to solve them, but besides the mechanics of how to solve those systems (or determine there are no solutions), linear algebra really came down to a study of vector spaces and linear transformations (structure-preseriving maps between vector spaces).

Abstract algebra, sometimes called modern algebra, fully embraces that concept with the study of sets endowed with one or more binary operations. Fields, like the real numbers, seem to have four operations (addition, multiplication, subtraction, division), although two of those four seem to be inverses to the other two. Rings like the integers have less structure (you can't divide two integers and always get an integer), and groups (a set with only one operation) like the permutations of the integers $\{1, 2, \dots, n\}$, are among the simplest objects we shall study. Of course their structural simplicity makes them ubiquitous and not surprisingly challenging to classify.

Abstract algebra is the study of groups, rings and fields, and like most of mathematics, tries to classify these objects "up to isomorphism", that is up to the point where they are algebraically indistinguishable. With vector spaces over a field, this was almost trivial since any two vector spaces over a field with the same dimension are isomorphic, so we need only one invariant to classify a vector space. For groups, rings, and fields the situation is more complicated, but also richer for the complication.