Suppose you were to run across two fields K < L. In a blatant attempt at foreshadowing instead of a field extension, we’ll refer to this pair as a “subfield.” Further let’s suppose that this “subfield” has “finite index”, namely that [L:K] is finite. As a representation theorist how might you go about studying this “subfield”?
A representation theorist studies a ring by studying the modules over it. So a representation theorist should study a pair of rings by studying the bimodules you can make from them. However, if we study all L-K bimodules than we’ve forgotten the inclusion structure. In particular if we had another inclusion of K into L we would still have the same category of bimodules. So instead let’s study the bimodules that we can make starting with the L-K bimodule which we get from the inclusion of K into L and the K-L bimodule .
What sort of operations can we perform on this bimodules? Well, we have a tensor product! So let’s consider the “tensor category” we get generated by and . (I put “tensor” in scare quotes here because we have two tensor products, tensor over K and tensor over L. So it isn’t a monoidal category, instead it’s what I like to call a bi-oidal category. More on this later.) What does this category C(L/K) look like?
First let’s decompose . This is isomorphic (using the multiplication map) to (where both actions come from the inclusion K<L). Clearly this breaks up into [L:K] copies of the trivial bimodule . Further tensoring these with anything else won’t do anything interesting.
So we’re halfway there, now let’s try decomposing . This is a bit trickier, let’s try an example. Let K be the real numbers and L the complex numbers. A basis over K for is given by , , , . We want to decompose this as an L-L bimodule.
A little computation yields the two following simple submodules:
- and . Notice that so this is isomorphic to
- and . Notice that so this is not isomorphic to . We denote it
Okay, what can we do with these new modules? Tensoring with will never do anything interesting. The only new computation is that
It is not difficult to see how this generalizes to an arbitrary Galois extension:
- The only K-K bimodule that occurs is . In any tensor product that lands in the K-K part of the category you can easily compute how many times this bimodule occurs by computing the K-dimension.
- The only L-K bimodule that occurs is . Again counting its multiplicity in tensor products is easy.
- The only K-L bimodule that occurs is . Again counting its multiplicity in tensor products is easy.
- The L-L bimodules are interesting. For every element we have the bimodule . Furthemore the tensor product decompositions are given by and
In summary, the Galois group appears naturally as the set of irreducible L-L bimodules that appear in C(L/K)!
Bonus questions: What if L/K is separable but non-Galois? What if L/K is totally inseparable?
(I’ve done examples in both cases, but they’re fun to look at yourself. I’ll answer questions in comments if people want hints or details.)