Frobenius homomorphism

Let F be a field of characteristicPlanetmathPlanetmath p>0. Then for any a,bF,

(a+b)p = ap+bp,
(ab)p = apbp.

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 automorphismPlanetmathPlanetmathPlanetmathPlanetmath, and is called the Frobenius automorphismMathworldPlanetmathPlanetmath.

Note: This morphismMathworldPlanetmath is sometimes also called the “small Frobenius” to distinguish it from the map aaq, with q=pn. 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