I am trying to figure out how to come up with field extensions of $\mathbb{Q}\_p$. Since abelian extensions are easy to write down over $\mathbb{Q}$, it seems natural to ask how to construct them for the p-adics. Given $\ell\neq p$ when is
$$
\mathbb{Q}\_p(\mu\_\ell)
$$
an abelian field extension?