Examples
We come back to the commutative examples of the definition in Section 7.3.
Usual arithmetic
Modular arithmetic
Polynomial rings
Residue class rings
The Gaussian numbers
Q is a subfield of R.
R is a subfield of C.
There are no proper subfields of Z/pZ (p prime).
For a subfield is also a subring
and we have seen in the examples of Section 7.1
that there are no proper subrings.
Q is a subring of the ring Q[X],
which is a field. One might speak of a subfield here, although the ring
Q[X] itself is not a field.
Later we shall see how to "extend" the domain Q[X] to a field.
Similar remarks hold for R and C instead of Q.
Let f = X4 - 2 
Q[X]
and consider
F = Q[X]/(f).
Since f is irreducible in Q[X],
this is a field.
See Section 4.5.
Now consider the element
b = X2 + (f) of F.
Q[X]
and consider
F = Q[X]/(f).
Since f is irreducible in Q[X],
this is a field.
See Section 4.5.
Now consider the element
b = X2 + (f) of F.
The subfield of F generated by Q and b
is K = Q + Qb.
(For instance, b2 = 2 and b-1 = b/2.)
Thus, the field F, which is a 4-dimensional vector space over Q
has a subfield K, which is a 2-dimensional linear space of F.
The field Q + Qi is a 2-dimensional subspace over Q.
An obvious subfield is Q. This is the only proper subfield of
Q + Qi, as will become clear later, from the fact that any subfield
contains Q.