# Galois group for the algebraic closure of a finite field

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.

• See this question for lack of elements of finite order. Abelian? How will a commutator act on any finite extension $E$ of $F$? Recall that $N$ is the union of such extensions $E$. – Jyrki Lahtonen Jun 16 '15 at 11:11