> started with rings and fields This seems unnecessary for a first class. I would agree with the parent here that the class should help build intuition and the

> started with rings and fields This seems unnecessary for a first class. I would agree with the parent here that the class should help build intuition and theoretical understanding. Presenting a rich theory with full generality is probably not the best way to do that though. There are some beautiful concepts and geometric understanding to be had with the reals. The abstraction to vector spaces defined over arbitrary rings and fields builds naturally once you have that base.