TITLE: Galois group of algebraic closure of a finite field
QUESTION [10 upvotes]: Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ?  

Thanks.

REPLY [6 votes]: No, there are not. Let $\overline{\mathbb{F}_p}$ be an algebraic closure of $\mathbb{F}_p$. Then $\overline{\mathbb{F}_p}/\mathbb{F}_p$ is an infinite Galois extension (infinite, algebraic, normal, separable) with pro-finite Galois group 
$$
Gal(\overline{\mathbb{F}_p}/\mathbb{F}_p)=\widehat{\mathbb{Z}}=\varprojlim \mathbb{Z}/n\mathbb{Z}.
$$
This is the pro-finite completion of $\mathbb{Z}$. By Artin-Schreier, the torsion subgroup
must be trivial or cyclic of order $2$. It is trivial, so there are no
no non-trivial elements of finite order. 
One could ask the same question for the absolute Galois group over $\mathbb{Q}$. Here it is cyclic of order $2$. 
The only element of finite order in the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ is, up to conjugation, complex conjugation.
For details on the group $\widehat{\mathbb{Z}}$ see Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers.