A field embedding of a field 
into a field 
is a field homomorphism 
that is an injection.
In particular, it preserves addition, multiplication,
and the multiplicative identity:
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
A field embedding 
is therefore sometimes simply defined as a nonzero field homomorphism from 
to 
. Here, nonzero
means not the zero homomorphism, i.e., not the map sending every element of 
to 0 in 
. Every nonzero field
homomorphism is an injection since its ring
kernel is an ideal in 
and the only ideals of a field 
are 
and 
.
If 
and 
are extension fields, an embedding of 
into 
over 
is a field embedding 
such that 
for every 
.
For a number field, embeddings into 
are classified as real embeddings
or imaginary embeddings. A number
field with no real embeddings is called a totally imaginary field.
