# Frobenius map

Let $K$ be any field of characteristic $p>0$, and suppose $K$ contains the finite field $\mathbb{F}_{q}$ of size $q$, where $q=p^{r}$. The $q^{\rm th}$ power Frobenius map on $K$ is the map $\operatorname{Frob}_{q}:K\longrightarrow K$ defined by $\operatorname{Frob}_{q}(x):=x^{q}$.

If $K$ is perfect, then $\operatorname{Frob}_{q}$ is an automorphism of $K$ which fixes $\mathbb{F}_{q}$, and accordingly is a member of the Galois group $\operatorname{Gal}(K/\mathbb{F}_{q})$.

Title Frobenius map FrobeniusMap 2013-03-22 12:34:52 2013-03-22 12:34:52 djao (24) djao (24) 6 djao (24) Definition msc 12E20 msc 11T99 FrobeniusAutomorphism