The Frobenius endomorphism (and the maps it induces on
cohomology) plays
a very important rôle in modern mathematics. Despite this
it is rather easily described.
Proposition Let K be a field of characteristic
p, for a prime number p. Then the function
F:K>K defined by F(a)=a^{p} is
a ring homomorphism. If K is a finite field
then F is an automorphism.
Definition
The ring homomorphism in the proposition is called the Frobenius endomorphism.
Before we prove the proposition we need:
Lemma The binomial coefficient ^{p}C_{n} is divisible
by p for 0<n<p.
Proof: ^{p}C_{n}=p!/n!(pn)!=p(p1)..(pn+1)/n!. Now
since n<p none of its prime factors divide p.
The same goes for all the other factors of n!. So
as the binomial coefficient is an integer n! must divide
into (p1)...(pn+1)
and we see that ^{p}C_{n} is divisible by p.
Proof of the proposition:

F(1)=1^{p}=1.

F(ab)=(ap)^{p}=a^{p}b^{p}=F(a)F(b).

F(a+b)=(a+b)^{p}=
a^{p}+pa^{p1}b+...+
^{p}C_{r}.a^{r}b^{pr}+...+pab^{p1}+b^{p}.
By the previous lemma all the binomial coefficients except the first and
last are divisible by p and hence zero in F.
It follows that
F(a+b)=a^{p}+b^{p}=F(a)+f(b).
Note that
F is always
injective
(this is true for any
ring homomorphism
from a field to itself).
Thus if
F is a finite field then by the
pigeonhole principle it must be a
bijection, as is required.
Its worth noting a special case of this. For the field Z_{p}
of integers modulo p the Frobenius is the identity map that fixes every element.
This follows by Fermat's little theorem. The same idea as the proof
of the theorem shows that in the ring Z_{p}[x]
the Frobenius is a homomorphism. Note that F(f(x))=f(x^{p}).