is a finite field of characteristic 
. Then 
is a subfield. We must have 
finite. It follows that has 
elements, because if 
is a 
-basis of every element of can be written uniqely in the form
1998/99 GCW
Finite Fields
A finite field clearly cannot have characteristic zero. Suppose is a finite field of characteristic . Then is a subfield. We must have finite. It follows that has elements, because if is a -basis of every element of can be written uniqely in the form
and we have 
choices for each 
.
The 
nonzero elements form a group under multiplication, so if 
is nonzero we have
whence every element of satisfies the equation
It follows that the polynomial
factorizes over as
So is the splitting field of 
. Hence for every prime 
and 
there is, up to isomorphism just one field with 
elements. It is called the Galois field 
.
Since
we deduce that in any commutative ring of characteristic 
the function 
taking 
to 
is a homomorphism. In particular, for 
it is an automorphism, satisfying 
. It is usually called the Frobenius automorphism.
Fact: 
is a cyclic group of order 
, with the Frobenius automorphism as generator.