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Think of it this way : by the local Kronecker-Weber, $K$ is the maximal 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 $K$.