Finite-Field Extensions are Cyclic Galois

Theorem.
Let $q=p^m$ be a prime power. For each $n\ge 1$, the extension

is a finite Galois extension . Its Galois group is cyclic of order $n$, generated by the Frobenius automorphism

which is the $m$-th iterate of the p-Frobenius .

Examples.

1. $\mathrm{Gal}(\mathbb{F}_{q^2}/\mathbb{F}_q)\cong C_2$, generated by $x\mapsto x^q$.
2. $\mathrm{Gal}(\mathbb{F}_{2^6}/\mathbb{F}_{2^2})$ has order $3$, generated by $x\mapsto x^{4}$.
3. Subfields of $\mathbb{F}_{q^n}$ correspond to divisors of $n$ via Galois correspondence .

Related. finite-field Galois group over F_p , existence/uniqueness of F_q .