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?
Best Answer
Perhaps you should separate this into at least two questions, as your first two questions are "local" and your last three are "global".
As to the local ones:
This response is more an idea than a complete answer. But it seems to me that the Krasner-Serre mass formula should tell you whatever you want to know about the number of extensions of a $p$-adic field of given degree. (Note that it comes down to counting totally ramified extensions, since any other kind is much easier to count.) Casting about just now for a good reference, I looked through my own course notes on local fields and was severely disappointed: I say too little and what I do say is riddled with typos. But this paper of Pauli and Roblot seems to be, among other things, a very thorough survey of these $p$-adic mass formulas. In particular it contains references to the original papers of Krasner (1966) and Serre (1978).
I haven't looked at the details myself, but surely (meaning, of course, that I am not completely sure!) this mass formula will answer your first question. It also seems to have a good chance to answer your second question, possibly along with some inclusion-exclusion/Mobius inversion arguments.
As to the global ones:
3) Yes! This is a famous theorem of Hermite. Look in a good algebraic number theory book, e.g. one written by Neukirch.
4) I don't, no, off the top of my head, but others surely do. Stay tuned...
5) I guess I don't see why this should be true, but I'll have to think more about it.
As above, asking fewer questions at a time will probably elicit more detailed answers.