  Let $\ell>2$ be a positive integer, $\zeta_\ell$ a primitive $\ell$-th root
of unity, and $K$ a number field containing $\zeta_\ell+\zeta_\ell^{-1}$ but
not $\zeta_\ell$. In a recent paper, Chonoles et. al. study iterated towers of
number fields over $K$ generated by the generalized Rikuna polynomial,
$r_n(x,t;\ell) \in K(t)[x]$. They note that when $K = \mathbb{Q}$, $t \in
\{0,1\}$, and $\ell=3$, the only ramified prime in the resulting tower is 3,
and they ask under what conditions is the number of ramified primes small. In
this paper, we apply a theorem of Gu\`ardia, Montes, and Nart to derive a
formula for the discriminant of $\mathbb{Q}(\theta)$ where $\theta$ is a root
of $r_n(x,t;3)$, answering the question of Chonoles et. al. in the case $K =
\mathbb{Q}$, $\ell=3$, and $t \in \mathbb{Z}$. In the latter half of the paper,
we identify some cases where the dynamics of $r_n(x,t;\ell)$ over finite fields
yields an explicit description of the decomposition of primes in these iterated
extensions.
