Determination of conductors from Galois module structure

Abstract.

Let E denote an unramified extension of ${{\bf Q}_p}$, and set $F = E({\zeta_{p^n}})$ for an odd prime p and $n \ge 1$. We determine the conductors of the Kummer extensions $F({\root p^n \of{a}})$ of F by those elements $a \in F{^{\times}}$ such that $F({\root p^n \of{a}})/E$ is Galois. This follows from a comparison of the Galois module structure of $F{^{\times}}$ with the unit filtration of F.