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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?

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The answer to 3 is yes, by the Hermite-Minkowski Theorem: there are only finitely many extensions of $\mathbb{Q}$ of degree $\leq n$, unramified outside $S$. –  Keenan Kidwell Apr 26 '11 at 14:11
    
@Keenan Kidwell Thanks for the reference. I suspected that it should be true. –  Andrei Jaikin Apr 26 '11 at 14:28
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What is a good reference which explaines why $G_{\mathbf{Q}_p}$ is topologically finitely generated? –  Hugo Chapdelaine Apr 26 '11 at 21:59
    
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3 Answers 3

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.

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@Pete L. Clark Thanks for the answer. I will look at the mass formula. –  Andrei Jaikin Apr 26 '11 at 16:13
    
A simple proof of the mass formula in the prime-degree case can be found in my $$ $$ Serre's "formule de masse" in prime degree (arxiv.org/abs/1005.2016) $$ $$ –  Chandan Singh Dalawat Apr 27 '11 at 2:49
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Re your question 3: Yes, for a given n there are only finitely many degree-n extensions of Q unramified outside S. Such an extension has discriminant bounded in terms of n and S and the number of degree-n extensions of bounded discriminant is finite, by an old theorem of Hermite. Unfortunately, the bounds obtainable by those methods for the number of such extensions are perhaps not as good as you would want: they would be exponential in n, at least (or at least so my mental arithmetic says.) But of course those bounds are likely to be very far from sharp.

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@JSE The exponential in $n$ is not bad. The subgroup growth of a finitely generated free profinite group is of type $(n!)^c$. –  Andrei Jaikin Apr 26 '11 at 16:16
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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.

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@Keenan Kidwell Thanks. –  Andrei Jaikin Apr 26 '11 at 16:17
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