Course description:

Abstract algebra grasps some essential common features of seemingly very different mathematical objects and provides a framework to investigate their structure in general. The topic of the course is an introduction to the theory of two such important algebraic structures: rings and groups. We will immediately realize that we already know many rings and groups of various forms from our earlier mathematical studies, and their unified treatment will provide a better understanding of their intrinsic properties. As a particularly interesting chapter, we will get to see how well the basics of number theory can be established in rings different from ℤ.

Topics:

Rings. Subrings, ideals, factor rings, homomorphism, direct sum. Number theory in rings: unique factorization theorem, principal ideal domains, Euclidean rings, Gaussian integers, two squares theorem. Field extensions, finite fields.

Groups. Subgroups, cyclic groups, order of an element. Cosets, Lagrange's theorem, Cauchy's theorem. Normal subgroups, conjugacy, factor groups, homomorphism, direct product. Permutation groups, Cayley's theorem, Burnside's lemma. Structure of groups of small size.