If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order.
If $N$ were a finite extension of $F$, then it would be easy to see that $\operatorname{Gal}(N/F)$ is a cyclic group. But I don't know if I can proceed using this idea in the above case.