A field homomorphism from a field 
to a field 
is a ring homomorphism 
. Under the usual convention
that homomorphisms preserve the multiplicative
identity, this means
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
If the zero homomorphism is allowed by convention, a nonzero field homomorphism is a field homomorphism that is not the map
sending every element of 
to 0 in 
. Every nonzero field homomorphism is nevertheless injective
since its kernel is an ideal
in 
and the only ideals of a field 
are 
and 
. Thus a nonzero field homomorphism has kernel 
and is a field
embedding.
