Why isn't this a counterexample to Local Kronecker-Weber? The local form of Kronecker-Weber tells me that every finite abelian extension of $\mathbb{Q}_p$ is contained in $\mathbb{Q}_p(\zeta_n)$ for some $n,$ where $\zeta_n^n =1.$
But what if $p$ is odd, and I have the finite abelian extension $\mathbb{Q}_p(\sqrt{p})$? Which $n$ could possibly work, seeing as this extension is ramified at $p$ but all these root of unity extensions are unramified?

The $p$-th root of unity extension (more generally, $p^k$-th root of unity) is ramified over $\mathbb{Q}_p$.
