## You are here

HomeFrobenius homomorphism

## Primary tabs

# 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*.

Defines:

Frobenius automorphism

Related:

FrobeniusMorphism, FrobeniusMap

Synonym:

Frobenius endomorphism, Frobenius map

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

12E99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

## Comments

## primality of p

Won't be a field if p isn't prime.

## Re: primality of p

thanks!

## Re: primality of p

(i had asked if p had to be prime)

## Re: primality of p

ahhh hehe, nice to know

since I don't see any problems with stating

"Let F be a field of characteristic p"

since F being prime that already implies p is prime (or zero)

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f