I have this exercise where I'm proving: "Every finite division ring is a field".
I need only a part (c) of it:
(a) show that a subalgebra of a finite dimensional central division algebra is a finite dimensional division algebra. (DONE)
(b) show that if $D$ is a finite dimensional central division algebra and $K\neq Z(D)$ is any subfield, then $D$ is generated as an $Z(D)$-algebra by $\bigcup_{d\in D^*}d^{-1}Kd$. (DONE)
(c) Conclude, without using the Noether-Skolem theorem, that a finite division ring is a field. (NEEDED…)
Thanks,
G.
Best Answer
To provide an alternate, maybe somewhat too over-loaded proof of this fact: every finite division ring is commutative.
It amounts to the same thing as showing that the Brauer group of any finite field is trivial, for then the finite division rings are all matrix rings. Since they are division rings, this implies that they are fields. Now, by a theorem in the theory of central simple algebras, $\mathbb {Br}(K/\kappa)$, for a finite Galois extension of a finite field $\kappa$, is isomorphic with $H^2(Gal(K/\kappa),K^\times)$. But finite extensions of finite fields are cyclic, so this is a cohomology group of a finite cyclic group. Since such cohomology is periodic of period $2$, we find that it is just the norm residue group $\kappa^\times/N_{\mathbb K\mid \kappa}\mathbb K^\times$. Since it is well-known that norms of finite field extensions are surjective, this tells us that the relative Brauer groups are trivial. As Brauer groups are colimits of relative ones, this finishes the proof.
P.S. I learned this proof from Mariano Suarez-Alvarez. As this proof is quite remarkable, I make it CW, to keep it as a reference. Thanks.