It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number theory. However, these books do not mention that this follows from the fact that the the absolute Galois group $G_{\mathbb Q_p}$ of $\mathbb Q_p$ is finitely generated. In fact, the structure of $G_{\mathbb Q_p}$ implies that there exists a constant $c$ such that the number of extensions of $\mathbb Q_p$ of degree $n$ is at most $c^n$. This is because $G_{\mathbb Q_p}$ is of exponential subgroup growth. So my first question is the following.

**Question 1:** Is there an alternative way to show that there exists a constant $c$ such that the number of extensions of $\mathbb Q_p$ of degree $n$ is at most $c^n$?

There is a method to show that if a profinite group $F$ does not have many subgroups, then it is finitely generated. For this consider $m_n(F)$ the number of maximal open subgroups of $F$ of index $n$. We say that $F$ is of polynomial maximal subgroup growth (PMSG) if there exists a constant $c$ such that $m_n(F)\le n^c$. We say that $F$ is positively finitely generated (PFG) if there exists $k$ such that $k$ random elements of $F$ generate $F$ with positive probability. In particular, if $F$ is PFG, then $F$ is finitely generated. A theorem of A. Mann shows that PMSG and PFG are equivalent. The group $G_{\mathbb Q_p}$ is prosoluble and finitely generated and so, by another result of A. Mann, $G_{\mathbb Q_p}$ is PMSG. So my second question is the following.

**Question 2**: Is there an alternative way to show that there exists a constant $c$ such that the number of minimal extensions of $\mathbb Q_p$ of degree $n$ is at most $ n^c$?(An extension $L/K$ is minimal if it does not contain proper subextensions)

The absolute Galois group of $\mathbb Q$ is not finitely generated. However we can look at the restricted ramification case. Let $S$ be a finite number of primes of $\mathbb Q$. Denote by $G_{\mathbb Q, S}$ the Galois group of the maximal extension of $\mathbb Q$ which is unramified outside the primes $S$. I believe that It is unknown whether this group is finitely generated. But what about the number of open subgroups of finite index?

**Question 3:** Is it true that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q$ of degree $n$ unramified outside the primes $S$?

if the answer on the previous question is yes.

**Question 4:** Do you know any upper bounds for the number of extensions of $\mathbb Q$ of degree $n$ unramified outside the primes $S$ and for the number of minimal extensions of $\mathbb Q$ of degree $n$ unramified outside the primes $S$?

Class field theory implies that for a given natural number $n$ there are only finite number of soluble Galois extensions of $\mathbb Q$ of degree $n$ unramified outside the primes $S$. This suggest the following question.

**Question 5:** Is the maximal pro-soluble quotient of $G_{\mathbb Q, S}$ finitely generated?