We describe a new algorithm, based on sieving procedures, for
determining the minimal index and all elements with minimal index in a
class of totally real quartic fields with Galois group $D_8$. It is
not universally applicable, but its applicability is easily checked
for any particular example, and it is very fast when applicable. We
include several tables demonstrating the potential of the method. (A
more general approach for quartic fields, described in [Gaál et al.],
requires much more computation time for each field.)
¶ Finally, we present a family of totally real quartic fields with
Galois group $D_8$ and having minimal index 1 (that is, a power
integral basis).