Frobenius homomorphism

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

${(a+b)}^p$ $=$ $a^p+b^p,$

${(ab)}^p$ $=$ $a^p{b}^p.$

Thus the map

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
Canonical name	FrobeniusHomomorphism
Date of creation	2013-03-22 12:22:50
Last modified on	2013-03-22 12:22:50
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	11
Author	mathcam (2727)
Entry type	Definition
Classification	msc 12E99
Synonym	Frobenius endomorphism
Synonym	Frobenius map
Related topic	FrobeniusMorphism
Related topic	FrobeniusMap
Defines	Frobenius automorphism