Frobenius mapFrobeniusMap2002-04-14 19:42:112002-12-02 07:08:19Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Frob}{\operatorname{Frob}}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 $\Frob_q: K \longrightarrow K$ defined by $\Frob_q(x) := x^q$.
If $K$ is perfect, then $\Frob_q$ is an automorphism of $K$ which fixes $\mathbb{F}_q$, and accordingly is a member of the Galois group $\operatorname{Gal}(K/\mathbb{F}_q)$.