Discriminants of simplest 3^n-tic extensions

Let $\mathrm{\ell }>2$ be a positive integer, ${\zeta }_{\mathrm{\ell }}$ a primitive $\mathrm{\ell }$-th root of unity, and $K$ a number field containing ${\zeta }_{\mathrm{\ell }}+{\zeta }_{\mathrm{\ell }}^{-1}$ but not ${\zeta }_{\mathrm{\ell }}$. In a recent paper, Chonoles et. al. study iterated towers of number fields over $K$ generated by the generalized Rikuna polynomial, $r_n(x,t;\mathrm{\ell })\in K(t)[x]$. 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 $r_n(x,t;\mathrm{\ell })$ over finite fields yields an explicit description of the decomposition of primes in these iterated extensions.