The Frobenius automorphism of a finite field 
of field
characteristic 
is the Frobenius map 
given by 
. By the freshman's
dream, this map is an injection: 
implies 
and hence 
(O'Sullivan 2006, p. 19). Since 
is finite, the map is also a
surjection, and so is a field
automorphism.
For a general field of field characteristic 
, the same Frobenius map
remains a ring homomorphism and an injection
but need not be a surjection.
