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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?

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On might view Galois theory being about representations of fields, no? – Jyrki Lahtonen Aug 24 '14 at 6:08
I wondered about that too. +1 – mick Dec 10 '14 at 22:18
up vote 7 down vote accepted

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.

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This is a pretty good answer, in fact one of the best I've read on here. I had a feeling it was something along these lines but couldn't quite put it in words. I guess by requiring that the map play well with the field operations ends up turning your vector space into a field (since you can multiply them by hypothesis). This then is just a question about field homomorphisms which I guess isn't the point anyway. Thanks for putting my vague ideas into concrete sentences! – Cameron Williams Aug 24 '14 at 5:19
Endomorphisms of an Abelian group form a ring, so one could reasonably ask for representations of rings in Abelian groups. I think something like this is going on in studying representations of associative algebras, though there's a vector space structure at both sides as well there. – Marc van Leeuwen Aug 24 '14 at 5:53
@MarcvanLeeuwen Yes! The thing is to have a useful representation theory and that usually comes from considering particularly nice (i.e., geometrically rich) structures). Of course, any group can be represented as a group of permutations on a set, but a set is not terribly rich. Similarly for representations of rings, as you say, just having an abelian group does not yield terribly interesting stuff. Having a vector space there helps. – Ittay Weiss Aug 24 '14 at 6:03
A small remark regarding these last comments, a $\mathbb{Z}$-algebra is a ring and a $\mathbb{Z}$-module is an abelian group, so it's just a special case, anyway. – Najib Idrissi Aug 24 '14 at 7:19

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