There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as .
It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!
Basically, continuity, or higher order conditions like differentiability seem to impose greater constraints on problems, which make them more solvable.
Some good examples of that:
- complex discrete problems:
- simple continuous problems:
- characterization of Lie groups
Besides the understandable Wikipedia definition, Video "Simple Groups - Abstract Algebra by Socratica (2018)" gives an understandable one:
Given a finite group and a simple group , find all groups such that is a normal subgroup of and .
We don't really know how to make up larger groups from smaller simple groups, which would complete the classification of finite groups:
In particular, this is hard because you can't just take the direct product of groups to retrieve the original group: Section "Relationship between the quotient group and direct products".