Could there be a noncommutative approach to the problem of counting field extensions?

Let \(K\) be a number field. I want to shortly discuss what appears to me as an amusing approach to the question of counting field extensions \(L|K\) of fixed degree \(d\) by discriminant. My main point is that such a counting should follow if we had a very good understanding (maybe including good knowledge of constants and error terms) of the three following counting problems:

1. How many central division algebras \(A\) of dimension \(d^2\) over \(K\) are there?
2. For a given field extension \(L \mid K\) of degree \(d\), how many central division \(K\)-algebras \(A\) contain \(L\) as a maximal subfield?
3. For a given central division algebra \(A\) of dimension \(d^2\), how many maximal subfields does it have?

Here, “how many” means “asymptotically as an upper bound on the discriminant grows to infinity”. To see how this can be used to count field extensions, notice that the number of pairs \((L,A)\), where \(L|K\) is an extension of degree \(d\) and \(A\) is a central division \(K\)-algebra of dimension \(d^2\) containing \(L\), can be obtained either by summing the answers to (3) for all answers to (1), or by summing the answers to (2) for all field extensions \(L|K\) of degree \(d\): in an unprecise way, the number of field extensions is “essentially” (1)*(3)/(2).

Question 1 was answered by Linowitz, McReynolds, Pollack and Thompson in their article Counting and effective rigidity in algebra and geometry (2018) (Theorem 1.5). A solution to question 2 is given in our recent preprint Asymptotics of extensions of simple ℚ-algebras (2024) (with Fabian Gundlach), building on group-theoretic results by Fein, Kantor and Schacher, and reproving the answer to question 1. (If the Galois group is fixed, the answer depends on \(L\) only up to a constant factor.)

Question 3 is certainly the hardest. (If we could take \(A = M_d(K)\), we would simply recover the original question, but the case where \(A\) is a division algebra seems more approachable.) For quadratic subfields of quaternion algebras (the case \(d=2\)), it has also been answered by Linowitz et al. (Theorem 1.6). The problem seems tightly related to the problem of determining “admissible groups”, which has been vastly studied by people like Schacher, Neftin, Harbater, Hartmann and Krashen. As I am interested in quantitative results, parametrizations are of course particularly important to me. For this, I think a promising approach is the use of Severi-Brauer varieties as “almost moduli spaces” for maximal subfields. This approach would be based on K-theory and the Merkurjev-Suslin theorem, leading to a parametrization of maximal subfields of a central division algebra \(A\) over \(K\) by elements of \(K_1(A) = K^{\times}\) (the last equality is due to Wang). A good reading on these subjects is the 5^th chapter of McFaddin’s PhD thesis.

Without seeing clearly how they fit, I think that the results of Chernousov, Rapinchuk, and Rapinchuk concerning division algebras having the same set of maximal subfields are also relevant to the discussion.

In some sense, this is nothing new and actually very classical, as it is essentially a continuation of the use of Hilbert symbols for classifying quadratic forms, and more generally of the use of Brauer groups in class field theory.

The approach I have been describing here is maybe flawed, doomed or hopeless for a reason which I am not seeing now. Or maybe it is indeed interesting but well-known. Do not hesitate to mail me any thoughts you may have concerning this silly idea!