Frobenius endomorphism

Definition. Let $K$ be a field of characteristic $p>0$. The Frobenius endomorphism is the map

It is a ring homomorphism because $(x+y)^p=x^p+y^p$ in characteristic $p$. For fields it is always injective; it is surjective exactly when $K$ is perfect .

On a finite field $\mathbb{F}_{p^n}$, Frobenius is an automorphism and generates the Galois group of $\mathbb{F}_{p^n}/\mathbb{F}_p$.

See also. field automorphism , finite field .

Examples.

1. On $\mathbb{F}_p$, Frobenius is the identity map since $x^p=x$ for all $x\in\mathbb{F}_p$.
2. On $\mathbb{F}_{p^n}$, the automorphisms fixing $\mathbb{F}_p$ are $\mathrm{Fr}^k:x\mapsto x^{p^k}$ for $k=0,\dots,n-1$.
3. On $K=\mathbb{F}_p(t)$, Frobenius is not surjective: there is no element $u\in K$ with $u^p=t$.