Why do we study representations of groups but not fields?

Question

Groups are great objects to work with as we all know. With surprisingly little structure, we can say fairly general things. However groups can be difficult to manage and so we look to representations to help simplify the matter. Group representations allow us to employ both the techniques from group theory as well as linear algebra. Fields are also great objects to work with. Perhaps it's a dumb question to ask, but why do we not study representations of fields as well? We now have two (abelian) groups to work with, which seems like it could complicate matters but on the surface it doesn't seem completely unreasonable. Is there some algebraic reason for why we shouldn't bother with representations of fields or even rings?

Answer 1

Groups are an abstraction of symmetry. A group representation is a way to realize the abstract symmetry encoded in a group by means of linear transformation, i.e., as geometric transformation of a particular nice nature of a linear space, often a finite dimensional one. With fields the situation is different. A field is not something that encodes symmetry (even though it has two groups associated with it). Moreover, to have a field representation on a linear space you need to somehow create a field from the linear space and then consider field homomorphisms $F\to Fieldification(V)$. There are two problems here. There is no natural way to turning a linear space into a field. And even if there were, then since field homomorphisms are all injective, the representation will just be an embedding of the field in the (non-existent) field-of-something-on-V.

In general, you can ask the same question about any two structures. Given structure $S_1$, why don't we consider representation of it using structure $S_2$. For this to make sense you must have a way of turning an $S_1$ structure into an $S_2$ structure by considering homomorphisms. It is unreasonable to expect this to be possible. The fact that endomorphisms in any category form a monoid and that automorphisms in any category form a group explains why you see representations of groups in various different places. But endomorphisms and automorphisms rarely form fields.