Minimal Conductors of Kummer Extensions by Roots of Unit Elements

We consider the question of finding conductors of degree pⁿ Kummer extensions of a finite extension F of Q_p(ζpⁿ). It appears to be a difficult question to attain a reasonable recipe for these conductors in terms of those elements α ϵ F^× with pⁿ th ischolar_main defining the Kummer extensions.