Galois group of a finite field extension is cyclic

Let $p$ be prime and let $\mathbb{F}_{p^n}$ be the finite field with $p^n$ elements.

Define the Frobenius map

\mathrm{Frob}_p:\mathbb{F}_{p^n}\to \mathbb{F}_{p^n},\quad x\mapsto x^p.

It is a field automorphism of $\mathbb{F}_{p^n}$ fixing $\mathbb{F}_p$ .

Theorem.

The extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is a finite Galois extension.
Its Galois group is cyclic of order $n$ : $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)=\langle \mathrm{Frob}_p\rangle \cong C_n.$ In particular, $\mathrm{Frob}_p^n=\mathrm{id}$ on $\mathbb{F}_{p^n}$ , and no smaller positive power is the identity.
More generally, if $m\mid n$ , then $\mathbb{F}_{p^n}/\mathbb{F}_{p^m}$ is Galois with $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})=\langle x\mapsto x^{p^m}\rangle \quad\text{and}\quad \big|\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})\big|=n/m.$

Remarks

Combined with the fundamental theorem of Galois theory, this implies that intermediate fields of $\mathbb{F}_{p^n}/\mathbb{F}_p$ correspond exactly to divisors of $n$ .

Examples

$\mathbb{F}_{p^2}/\mathbb{F}_p$ . The Galois group has order $2$ , generated by $x\mapsto x^p$ . This is the unique nontrivial $\mathbb{F}_p$ -automorphism of $\mathbb{F}_{p^2}$ .

$\mathbb{F}_{2^3}/\mathbb{F}_2$ . The Galois group is cyclic of order $3$ , generated by $x\mapsto x^2$ . The intermediate fields correspond to divisors of $3$ , so there are no proper intermediate fields besides $\mathbb{F}_2$ .

$\mathbb{F}_{p^6}/\mathbb{F}_{p^2}$ . Since $2\mid 6$ , this is Galois of degree $3$ and $\mathrm{Gal}(\mathbb{F}_{p^6}/\mathbb{F}_{p^2})=\langle x\mapsto x^{p^2}\rangle\cong C_3.$