Finite-field Frobenius automorphism ($x\mapsto x^q$)

The map $x\mapsto x^q$ is an automorphism of ${\mathbb{F}}_{q^n}$ fixing ${\mathbb{F}}_q$. Its powers are all the ${\mathbb{F}}_q$-automorphisms, and its order is $n$.

Over ${\mathbb{F}}_q$, the roots of the minimal polynomial of $\alpha \in {\mathbb{F}}_{q^n}$ are its distinct iterates $\alpha ,{\alpha }^q,{\alpha }^{q^2},\dots$ under the finite-field Frobenius automorphism.

Every finite field extension is a Galois extension, and