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# 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*

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## 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