R = Z
R = Z + Zi where i = 
-1.
(The Gaussian integers)
R = Q[X]
Q(Z) = Q.
In fact,
starting from the integers,
one would implement the rational numbers on computer
in just the way the fraction field is constructed,
see the application of the theorem.
Q(Z + Zi) = Q + Qi.
For
(a + bi)/(c + di) =
(ac - bd)/(c2 + d2) +
((ad + bc)/(c2 + d2))i.
Q(Q[X]) = Q(X),
the so-called rational functions field in X.
Q(X) = {f/g | f, g
L[X] and g(X)
0}.
This field can be described in much the same way as in the theorem on page 4 of Section 7.3:
L[X] and g(X)
0}.