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)
|
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(2)
|
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(3)
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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.
