Is there a connection between Artin-Schreier theorem on finite groups which can be absolute Galois groups and the classification of finite groups freely acting on even-dimensional sphere?
The former says that all finite extensions $[E:F]<\infty$ with $E$ algebraically closed are $E=F[\sqrt {-1}]$, so the only finite absolute Galois group is $\mathbb Z_2$.
The latter says that the only finite group that can freely act on even-dimensional sphere is also $\mathbb Z_2$, by the Lefschetz fixed point theorem.
An idea.
If $E/F$ is a finite extension, then we can construct the projective space $P_FE$ corresponding to $E$ viewed as an $F$-vector space. The multiplication on $E$ induces an abelian group structure on $P_FE$, which turns it into an algebraic group defined over $F$.
I seriouly doubt there are many abelian group structures on projective spaces (this should be proved by looking at the cohomology) so this has to restrict at least the degree of the extension. After this, we are left with a projective space with dimeension in a very small set and on which a finite group acts in a rather special way, and hand waving will show that the order of the group is $2$.
Remark. If we take $F=\mathbb R$, then $P_FE$ is a real projective space and a compact abelian real Lie group, and we can consider them as smooth manifolds. The latter means it is a real torus, and as no positive dimensional projective space is a a torus (this is can be checkd looking at the de Rham cohomology of these manifolds) we conclude that $\dim P_FE=0$, that is, that $\dim_FE=2$. The extension is therefore quadratic, and we did not need any fixed point theorem.