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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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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 – Chandan Singh Dalawat Nov 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). – René Nov 4 '12 at 12:03
It is simpler in your situation to consult Section 2 of – Chandan Singh Dalawat Nov 4 '12 at 12:18
References to the literature seem like overkill. This question needs nothing but Galois theory and basic properties of $p$-adic fields. – René Nov 4 '12 at 12:24

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