Discriminants of simplest $3^n$-tic extensions

Abstract

Let ℓ > 2 be a positive integer, ζ ℓ a primitive ℓ-th root of unity, and K a number field containing ζ ℓ + ζ −1 ℓ but not ζ ℓ . In a recent paper, Chonoles et. al. study iterated towers of number fields over K generated by the generalized Rikuna polynomial, rn(x, t; ℓ) ∈ K(t) [x]. They note that when K = Q, t ∈ {0, 1}, and ℓ = 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 Guàrdia, Montes, and Nart to derive a formula for the discriminant of Q(θ) where θ is a root of rn(x, t; 3), answering the question of Chonoles et. al. in the case K = Q, ℓ = 3, and t ∈ Z. In the latter half of the paper, we identify some cases where the dynamics of rn(x, t; ℓ) over finite fields yields an explicit description of the decomposition of primes in these iterated extensions.Date: July 12, 2018.