Galois Group of a Finite Field Extension is Cyclic

Theorem.
Let $q=p^n$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. The extension $\mathbb{F}_q/\mathbb{F}_p$ is Galois , and

is cyclic of order $n$, generated by the Frobenius map $\varphi(x)=x^p$.

More generally, $\mathrm{Gal}(\mathbb{F}_{q^m}/\mathbb{F}_q)$ is cyclic of order $m$, generated by $x\mapsto x^q$ (see finite-field Galois cyclicity ).

Examples.

1. $\mathrm{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)=\{1,\varphi\}$ where $\varphi(x)=x^p$.
2. $\mathrm{Gal}(\mathbb{F}_{2^3}/\mathbb{F}_2)$ is cyclic of order $3$; its elements are $1,\varphi,\varphi^2$.
3. For $\mathbb{F}_{q^6}/\mathbb{F}_q$, the Galois group is cyclic of order $6$, generated by $x\mapsto x^q$.

Related. Galois groups , Galois correspondence .