Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the
field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ `
is abelian over $\mathbb{Q}_{p}$?

First try the case $n=1$. Try to find a criterion for a degree-$p$ cyclic extension $L$ of $K={\mathbf Q}_p(\zeta_p)$ to be galoisian (resp. abelian) over $F={\mathbf Q}_p$, or extract such a criterion from Section 4 of arxiv.org/abs/1005.2016.
– Chandan Singh DalawatNov 4 '12 at 11:50

Any subextension of an abelian extension is Galois. Now try to find a non-Galois subextension in your case (with $u$ not being a $p^n$th power).
– RenéNov 4 '12 at 12:03

References to the literature seem like overkill. This question needs nothing but Galois theory and basic properties of $p$-adic fields.
– RenéNov 4 '12 at 12:24