Let F be a field. The following definitions are completely standard (compare them with those for monoids, groups, and rings given so far).
A subfield of F is a subring of F which is closed under inverses of nonzero elements.
If X is a subset of F, the subfield of F generated by X is the intersection of all subfields containing X.
We now focus on subfields of the field of complex numbers.
Let K be a subfield of C. If a is an element
of K, and f, g are polynomials in
Q[X], then f(a)/g(a) is an
element of K whenever g(a) 
0. The set of all these fractions
makes up the smallest subfield of C that contains a.
Of course, instead of polynomials in Q[X], we could have chosen
f and g with coefficients in any subfield L of K.
In general we obtain the following.
If a 
C and
L a subfield of C, then
L[X] and g(a)
0}
is the subfield of C generated by a and L.