If $K=\langle$HCF of $\mathbb{Q}_{p}$, $\mathbb{Q}^\mathrm{cycl}\rangle$, does $K$ also contain all roots of elements of $\mathcal{O}_{K}^{\times}$?

Question

Suppose $K$ is an extension of $\mathbb{Q}_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the Hilbert Class Field (i.e., maximal unramified extension) of $\mathbb{Q}_{2}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of all elements of $\mathcal{O}_{K}^{\times}$?

This seems to boil down to knowing the ramification behavior of extensions obtained by adjoining roots of units in $\mathcal{O}_{K}^{\times}$. I feel like this should be hidden somewhere in the main results of class field theory, but I'm not sure.

Answer 1

Think of it this way : $K$ is an abelian extension of $\mathbf{Q}_p$. Now, the the extension $\mathbf{Q}_p(\root{p^m}\of u)$ need not even be galoisian over $\mathbf{Q}_p$ for some appropriate $u\in\mathbf{Z}_p^\times$, so it cannot be contained in $K$. (As it happens, $K$ is the maximal abelian extension of $\mathbf{Q}_p$ by the local Kronecker-Weber theorem, but this fact is not used in the above argument.)