Frobenius automorphism of a finite field

Definition

Let $F$ be a finite field of characteristic $p$. The Frobenius automorphism of $F$ is the map $\varphi \colon F \to F$, $\varphi(x) = x^p$.

Remarks

The map $\varphi$ is the Frobenius endomorphism of $F$. It is injective, as is every field homomorphism, so it is bijective, because $F$ is finite. The map $\varphi$ fixes every element of the prime subfield $\mathbb{F}_p$, by Fermat's little theorem. If $|F| = p^n$, then $\varphi^n$ is the identity, and the automorphisms of $F$ are exactly $\mathrm{id}, \varphi, \dots, \varphi^{n-1}$. So $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by $\varphi$.

Explanations

L1 human

Raising to the power $p$ respects addition in characteristic $p$, by the freshman's dream. In a finite field, this map permutes the elements, and every symmetry of the field is a power of it.