  For each real quadratic field we constructively show the existence of
infinitely many exceptional quartic number fields containing that quadratic
field. On the other hand, another infinite collection of quartic exceptional
fields without any quadratic subfields is also provided. Both these families
are non-Galois extensions of $\mathbf{Q}$, and their normal closu res have
Galois groups $D_4$ and $S_4$ respectively. We also show that an infinite
number of these exceptional quartic fields have power integral basis, i.e.,
monogenic. We also construct large collections of exceptional number fields in
all degrees greater than 4.
