Often one studies finite or more generally algebraic field extensions. It is natural to wonder whether familiar properties hold when these conditions are removed; or, as things arise in practice, whether one can make the appropriate definitions so that familiar properties hold for these extensions with respect to the generalized definitions. In particular, we will talk in detail about extending Galois theory to the infinite case. Depending on time, we will talk about an algebraic notion of a basis for transcendental (i.e. non-algebraic) extensions (note that this notion of a basis will differ from that of infinite vector spaces, because we want to encode transcendentality).
(The talk should be accessible to everybody -- elementary algebra required).