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 $\mathbb{Q}_{p}$?
0

$\begingroup$ 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. $\endgroup$ – Chandan Singh Dalawat Nov 4 '12 at 11:50

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

$\begingroup$ It is simpler in your situation to consult Section 2 of arxiv.org/abs/0912.2829. $\endgroup$ – Chandan Singh Dalawat Nov 4 '12 at 12:18

1$\begingroup$ References to the literature seem like overkill. This question needs nothing but Galois theory and basic properties of $p$adic fields. $\endgroup$ – RP_ Nov 4 '12 at 12:24