Course overview
The algebraic notions of groups and rings are of great interest in their own right, but knowledge and understanding of them is of benefit well beyond the realms of pure algebra. Areas of application include, for example, advanced number theory; cryptography; coding theory; differential, finite and algebraic geometry; algebraic topology; representation theory and harmonic analysis including Fourier series. The theory also has many practical applications including, for example, to the structure of molecules, crystallography and elementary particle physics.
Topics covered are: (1) Groups, subgroups, cosets and normal subgroups, homomorphisms and factor groups, products of groups, finitely generated abelian groups, groups acting on sets and the Sylow theorems. (2) Rings, integral domains and fields, polynomials, ideals, factorization in integral domains and unique factorization domains.
Course learning outcomes
- Demonstrate understanding of the idea of a group, a ring and an integral domain, and be aware of examples of these structures in mathematics.
- Appreciate and be able to prove the basic results of group theory and ring theory.
- Understand and be able to apply more advanced results on groups: the fundamental theorem of finitely generated abelian groups, Burnside's theorem and the Sylow theorems.
- Appreciate the significance of unique factorization in rings and integral domains.
- Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
- Demonstrate skills in communicating mathematics orally and in writing.