Question

A goal which I have been pursuing is to understand how number fields are distributed with respects to their invariants. To be more precise I was captivated by the following question: Let $N(X,n,G)$ be the number of number fields of dimension $n$ where $G$ is the Galois group of its Galois closure, and their discriminants is bounded by $|X|$ up to isomorphism. This question might be natural to ask: For which group $G\leq S_n$ one might get a positive proportion of number fields when $X\to \infty$.

From Class Field Theory or one can show it more elementary, using Delone-Faddeev correspondence, for $n=3$, $C_3$ has a density 0. And also when $G$ is an abelian group then the answer would be same as $C_3$ by using class field theory. Prof. Manjul Bhargava proved for $n=4$ , $D_4$ has a positive density, and also he showed for $n=5$ no other groups, except $S_5$, could have positive density.

I think, once Manjul told me, one might expect for $n=p$, $p$ is a prime number, the only group which can contribute with positive density is $S_p$. But I could not even find a heuristic that why this should be true.

Is there a heuristic or even a theorem which can support the above expectation? Or more generally what do we know about $N(X,n,G)$?

Answer 1

There are general conjectures and heuristics; see for example a recent survey by Ellenberg and Venkatesh on this and the function field analog.

Roughly, there is a (modified) conjecture of Malle asserting that given $G$ the number of respective extensions of discriminant at most $X$ is asymptotically \[ c x^{1/a} (\log x)^b \] with a specific dependence of $a,b$ on $G$; and there are some heuristics of Bhargava for $c$. The 'modified' is due to the fact that nowadays conjecture is not exactly the original version of Malle (Klüners gave a counter example to the original some years ago).

This circle of ideas is also linked to the Cohen--Lenstra heuristics on class groups; this is also discussed in this survey.

Answer 2

A conjecture of Malle (which is known to be false in general, but probably still provides a good heuristic in many cases) implies that if $G$ is a transitive subgroup of $S_n$, then a positive proportion of number fields (ordered by discriminant) of degree $n$ have Galois group $G$ if and only if $G$ contains a transposition. The relevant constant here is $a$ from unknown (google)'s answer: it's the minimum value of $n$- the number of cycles in $g$, where $g$ runs over all non-identity elements of $G$. For example, when $n=4$, the dihedral group $D_4$ contains a transposition, which explains why a positive proportion (roughly 11%) of quartic fields have Galois group $D_4$.

See this paper by Jürgen Klüners for the precise statement of the conjecture, together with a counterexample.

Answer 3

First of all, the most general conjecture about N(X,n,G) is indeed due to Malle, and there are indeed counterexamples to the original formulation due to Kluners. Seyfi Turkelli has a very nice paper which explains "why" those counterexamples arise by means of a function field analogy, and offers a revised version of Malle's conjecture which seems to me pretty solid.

Why does only S_p give positive density? Because under Malle's conjecture, N(G,n,X) will be asymptotic to X^a(G) * some power of log, where a(G) is at most 1, with equality only when G contains a transposition. So the class of groups expected to have positive density is precisely those containing a transposition. When p is prime, the only transitive subgroup of S_p containing a transposition is S_p itself, so you're done. On the other hand, both D_4 and S_4 are transitive subgroups of S_4 containing a transposition -- indeed, they are the only such, so they're the only two Galois groups that are supposed to arise for a positive density of quartic fields ordered by discriminant.