When they have trouble understanding a theorem, ordinary mathematicians ask: "What's an example of this?"

Category theorists ask: "What's this an example of?"

(1/n)

I'm in that situation myself trying to learn about division algebras and how they're connected to Galois theory. Gille and Szamuely's book "Central Simple Algebras and Galois Cohomology" is a great introduction, and right now it's free here:

http://www.math.ens.fr/~benoist/refs/Gille-Szamuely.pdf

(2/n)

But one of the key ideas, "Galois descent", was explained in a way that was hard for me to understand.

It was hard because I sensed a beautiful general construction buried under distracting details. Like a skier buried under an avalanche, I wanted to dig it out.

(3/n)

I started digging, and soon saw the outlines of the body.

We have a field k and a Galois extension K. We have the category of algebras over k, Alg(k), and the category of algebras over K, Alg(K). There is a functor

F: Alg(k) -> Alg(K),

a left adjoint.

(4/n)

We fix A ∈ Alg(K). We want to classify, up to isomorphism, all a ∈ Alg(k) such that F(a) ≅ Alg(K). This is the problem!

The answer is: the set of isomorphism classes of such a is

H¹(Gal(K|k), Aut(A))

Here H¹ is group cohomology, and Gal(K|k) is the Galois group.

(5/n)

The group Gal(K|k) acts on Aut(A), which is the automorphism group of A.

Whenever you have a group G acting on a group K, there's a set you can define, called the "first cohomology set", H¹(G,K). This set is the answer to our problem when G = Gal(K|k), K = Aut(A).

(6/n)