  A standard formula (1) leads to a proof of HT90, but requires proving the
existence of $\theta$ such that $\alpha\ne 0$, so that
$\beta=\alpha/\sigma(\alpha)$. We instead impose the condition (M), that taking
$\theta=1$ makes $\alpha=0$. Taking $n=3$, we recover Shanks's simplest cubic
fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic
quartic fields, and a certain family of totally real cyclic extensions of
$\Q(\cos(\pi/4m))$ all have defining polynomials whose zeroes satisfy
\eqref{e:M}. Further investigation of (M) for $n=4$ leads to an elementary
algebraic construction of a 2-parameter family of octic polynomials with
"generic" Galois group $_{8}T_{11}$. Imposing an additional algebraic condition
on these octics produces a new family of cyclic quartic extensions. This family
includes the "simplest" quartic fields and Washington's cyclic quartic fields
as special cases. We obtain more detailed results on our octics when the
parameters are algebraic integers in a number field. In particular, we identify
certain sets of special units, including exceptional sequences of 3 units, and
give some of their properties.
