It is well known that there is a parallel between Galois theory and covering theory. So I wonder whether there is a deep similarity between the following two facts:

**Artin-Schreier theorem.** The only non-trivial finite field extensions $U \supset V$ with $U$ an algebraically closed field are $U=V(\sqrt {-1}),$ so among non-trivial finite groups only $\mathbb Z/2\mathbb Z$ may "Galois-act" on an algebraically closed field.

**Corollary of Lefschetz fixed point theorem**. The only non-trivial finite coverings $U \to V$ with $U$ an *even-dimensional sphere* are given by antipodal maps, so among non-trivial finite groups only $\mathbb Z/2\mathbb Z$ may act freely on (these particular) spaces that are the universal coverings of themselves.

Unfortunately, the statement from covering theory is very particular. Maybe, there is a simple case of the first statement where the similarity may be seen even better?