An Enhancement to the Galois Correspondence in Abelian Extensions of Q

Algebra is a study of mathematics geared towards structure and classification. When you think of the three core facets of modern abstract algebra, group, ring and field theory, each can seem somewhat unrelated past a surface level.

For instance, most introductory courses on group theory focus on finite groups, symmetry, and finding subgroups of given groups. Most courses on ring theory focus on infinite rings, ideals, and subrings. Field theory on the other hand has a different focus entirely. It’s mainly about field extensions, rather than subfields. When you learn each of these theories, they certainly have common points, but each feels unique in their focus. So what happens when two of these completely different branches of algebra collide?

Group theory and field theory are connected in a way seemingly unimaginable at first, and this beautiful correspon- dence is the core of Galois theory. However, it does not end at the Galois theory you would learn in a typical abstract algebra course. In certain abelian Galois extensions, there is an enhancement to the normal correspondence. The aim of this talk is to explore that enhanced correspondence.

We want this talk to be accessible to anyone, and as such, will be spending some time in the beginning explaining background information for those with minimal algebra experience. Please feel free to ask any questions you have. Also, if you have an idea of where the talk might be going, please speak up and join in!