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.
