Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. Put $K_\infty = K(\mu_{p^\infty})$, the field extension obtained by adjoining all $p$-power roots of unity to $K$.

I want to prove that: if $u \in {\mathcal{O}_K^\times}$ is not a root of unity, then the field extension $K_u$ of $K_\infty$ generated by the $p$-power roots of $u$ is non-abelian over $K$.

Here is a vague idea that I believe will allow us to prove the statement:

First, note that $\mathcal{O}_K^\times = F \times (1 + m_K)$, where $F$ is a finite group and $1 + m_K$ is the group of principal units. We may then reduce the problem to the case when $u$ is a principal unit. Kummer theory provides us with an isomorphism $$ K^\times / (K^\times)^{p^n} \cong H^1 (G_K, \mu_{p^n}), $$ for all $n \in \mathbb{Z}_{\geq 1}$. Taking the projective limits for $n$, we get $$\widehat{K^\times} \cong H^1 (G_K, \mathbb{Z}_p(1)).$$ We consider $u$ now as an element of $\widehat{K^\times}$. I saw in a recent paper by Khare and Wintenberger entitled, Ramification in Iwasawa theory and Splitting Conjectures, that

- $K_u$ is the extension of $K_\infty$ corresponding to the fixed field of the kernel of the homomorphism arising from the image of $u$ under the map $$ f: \widehat{K^\times} \cong H^1 (G_K, \mathbb{Z}_p(1)) \rightarrow \text{Hom} (G_{K_\infty}, \mathbb{Z}_p)(1)^{\text{Gal}(K_\infty/K)}.$$

Let $\rho = f(u)$. Then, to prove that $K_u$ is non-abelian over $K$, it is enough to verify that the image of $\rho$ is non-abelian.

Now, with this line of argument,

(1) I do not know how the assumption that $u$ is a principal unit and is not a root of unity comes to use;

(2) I am having difficulty in understanding the statement in bullet, so I would be glad if someone can explain to me how $f$ was constructed and how come $K_u$ can be interpreted in the manner described above;

(3) I wonder if there is some effect to the proof if $K$ is assumed to contain some $p$-power roots of unity.

Of course, I also welcome other ideas that can lead to the proof of the above statement. Thanks in advance.