# Frobenius homomorphism

Let $F$ be a field of characteristic $p>0$. Then for any $a,b\in F$,

 $\displaystyle(a+b)^{p}$ $\displaystyle=$ $\displaystyle a^{p}+b^{p},$ $\displaystyle(ab)^{p}$ $\displaystyle=$ $\displaystyle a^{p}b^{p}.$

Thus the map

 $\begin{matrix}\phi:F&\to&F\\ a&\mapsto&a^{p}\end{matrix}$

is a field homomorphism, called the Frobenius homomorphism, or simply the Frobenius map on $F$. If it is surjective then it is an automorphism, and is called the Frobenius automorphism.

Note: This morphism is sometimes also called the “small Frobenius” to distinguish it from the map $a\mapsto a^{q}$, with $q=p^{n}$. This map is then also referred to as the “big Frobenius” or the “power Frobenius map”.

Title Frobenius homomorphism FrobeniusHomomorphism 2013-03-22 12:22:50 2013-03-22 12:22:50 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 12E99 Frobenius endomorphism Frobenius map FrobeniusMorphism FrobeniusMap Frobenius automorphism