TITLE: Finitely generated Galois groups
QUESTION [16 upvotes]: 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?

REPLY [4 votes]: Regarding the local question, you might be interested in John Jones' website http://hobbes.la.asu.edu/research.html. He has a database where he gives all sorts of data for all the extensions of $\mathbb{Q}_p$ of degree at least through $8$ (polynomials that give the extensions, etc.). His interactive table may go up to degree 12 now (this was the problem his most recent thesis student was working on). Anyway, he has a number of papers on the methods used to compute all the extensions of a given degree, and the references therein may be useful for answering your question (for example I remember finding the mass formula that Pete mentions as well as the Pauli and Roblot paper among them). The main paper about the database is here.