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 $\mathbb{Q}_{p}$?
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$\begingroup$ 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 DalawatNov 4, 2012 at 11:50
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$\begingroup$ 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). $\endgroup$– R.P.Nov 4, 2012 at 12:03
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$\begingroup$ It is simpler in your situation to consult Section 2 of arxiv.org/abs/0912.2829. $\endgroup$– Chandan Singh DalawatNov 4, 2012 at 12:18
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1$\begingroup$ References to the literature seem like overkill. This question needs nothing but Galois theory and basic properties of $p$-adic fields. $\endgroup$– R.P.Nov 4, 2012 at 12:24
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