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
Canonical name	FrobeniusMap
Date of creation	2013-03-22 12:34:52
Last modified on	2013-03-22 12:34:52
Owner	djao (24)
Last modified by	djao (24)
Numerical id	6
Author	djao (24)
Entry type	Definition
Classification	msc 12E20
Classification	msc 11T99
Related topic	FrobeniusAutomorphism