This an exercise in Algebraic Number Theory written by Jürgen Neukirch. It is in chapter $2$, section $5$. The question is as follows:
For a $\mathfrak{p}$-adic number field $K$, every subgroup of finite index in $K^*$ is both open and closed.
Here is what I thought:
I think that we may use proposition $(5.7)$ on page $140$, if we denote the subgroup as $G$, then there is a subgroup corresponding to $G$ in $\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}/p^\alpha\mathbb{Z}\oplus\mathbb{Z}_p^d$ with finite index too, as $Z$, $Z/(q-1)Z$, and $Z/p^\alpha Z$ have discrete topology, and the only subgroups with finite index of $Z_p$ are of the form $p^n Z_p$ for some integer n, which is both open and closed in $Z_p$, so I think we may use these facts to solve this question. I am not sure if this is right, so I hope we can discuss it and I could learn more. Thank you for seeing and answering it.
Here is the proposition $(5.7)$ on page $140$ in Algebraic Number Theory written by Jürgen Neukirch:
$(5.7)$ Proposition: Let $K$ be a local field and $q = p^f$ the number of elements in the residue class field. Then the following hold.
(i) If $K$ has characteristic $0$, then one has (both algebraically and topologically) $K^*\cong\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}/p^\alpha\mathbb{Z}\oplus\mathbb{Z}_p^d$ $\qquad$ where $a>=0$ and $d$=[$K$:$Q_p$].
(ii) If K has characteristic p, then one has (both algebraically and topologically) $ K^*\cong\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}_p^N $
Actually my basic idea is to analyze the same topological structure problem on a homeomorphism but easier algebraic group as $\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}/p^\alpha\mathbb{Z}\oplus\mathbb{Z}_p^d$ or $\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}_p^N $. And we know $Z$, $Z/(q-1)Z$ and $Z/p^\alpha Z$ all have discrete topology and their subgroups are open and closed, since the quotient group is discrete. To $Z_p$, its only subgroup of finite index is of the form $p^nZ_p$ for some $n$, and this subgroup is both open and closed in $Z_p$, so every subgroup with finite index of $\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}/p^\alpha\mathbb{Z}\oplus\mathbb{Z}_p^d$ or $\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}\oplus\mathbb{Z}_p^N $ is both open and closed. So every subgroup with finite index of $K^*$ is both open and closed.