I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me.


Let $F$ be a finite extension of $\mathbb{Q}\_\ell$. Let $F\_1$ is a finite Galois extension of $F$ and $F\_\infty$ is a Galois extension of $F$ which is also an extension of $F\_1$. Suppose we know $\mathrm{Gal}(F\_\infty/F\_1)\cong \mathbb{Z}\_p$ or $\mathbb{Z}\_p\times \mathbb{Z}\_p$ for some $p\neq \ell$. With these assumption, why the extension $F\_\infty/F\_1$ should be unramified? The only information I know is the result comes from class field theory, but I cannot catch it.



Unramified extension and class field theory
Hint: By class field theory, ramified abelian extensions of $F\_1$ correspond to subgroups of the unit group of the ring of integers of $\mathcal{O}\_{F\_1}$, which is isomorphic to $\mu(F\_1) \times \mathbf{Z}/\ell^a \times \mathbf{Z}\_\ell^d$, a finite group times a pro-$\ell$ group.


