Resultant valuation for prime-power periods

Prove that for every prime number p greater than or equal to 3 and every integer r greater than or equal to 1, with n=p^r, the 2? p-adic valuation of the resultant α_n=Res(Φ_n,a_n) satisfies v_p(α_n)=φ(n)/(p−1)=p^{r−1}, and that the normalized resultant α_n/p^{v_p(α_n)} is congruent to 1 modulo n.

Background

The paper studies the quadratic family F_λ(z)=λz(1−z) and the integer resultants α_n=Res(Φ_n,a_n), where Φ_n is the nth cyclotomic polynomial and a_n is obtained from the coefficients of the formal linearizing map. The p-adic valuation v_p(α_n) measures the divisibility of this resultant by a prime p.

Computations suggest that when n is a multiple of a prime p≥3, the expected valuation is φ(n)/(p−1), although exceptions occur. The authors state that these exceptions appear not to occur when n is a power of p, and formulate the prime-power valuation and congruence as a conjecture. The conjecture is proved in the paper only in the case r=1.