ON THE DISTRIBUTION OF GALOIS GROUPS, II

GUNTER MALLE

Abstract. We propose a very precise conjecture on the asymptotics of the counting function for extensions of number fields with fixed Galois group and bounded norm of the discriminant. This sharpens a previous conjecture of the author. The conjecture is known to hold for abelian groups and a few non-abelian ones. We give a heuristic argument why the conjecture should by true. We also present some computational data for the non-solvable groups of degree 5.

1. Introduction

We are interested in the distribution of number fields with a given Galois group. We propose a very precise conjecture on the density of fields having given Galois group and bounded discriminant.

Let G be a finite transitive permutation group on n points and let k be a number field. By abuse of notation we will write Gal(K/k) = G if K/k is a field extension such the Galois group of the Galois closure K̂/k viewed as permutation group on the set of embeddings of K into K̂ is permutation isomorphic to G. Our goal is the description of the asymptotic behaviour of

Z(k, G; x) := | {K/k | Gal(K/k) = G, N_{k/ℚ}(d_{K/k}) ≤ x} |,

the number of field extensions of k (inside a fixed algebraic closure ℚ̄) of degree n with Galois group isomorphic to G and norm of the discriminant d_{K/k} bounded above by x.

We propose the following conjecture on the growth of Z(k, G; x) as x goes to infinity, that is, on the density of Galois groups:

Conjecture 1.1. Let G ≠ 1 be a transitive permutation group and k a number field. Then there exists a constant c(k, G) > 0 such that

Z(k, G; x) ∼ c(k, G) x^{a(G)}(log x)^{b(k,G)−1},

for certain explicit constants a(G), b(k, G) (see Definitions (1) and (2)).

Here, for functions f, g : ℝ → ℝ_{>0} we write f ∼ g if

lim_{x→∞} \frac{f(x)}{g(x)} = 1.

2000 Mathematics Subject Classification. 11R32, 11R29, 12-04.

Key words and phrases. Galois groups, density of extensions, distribution of discriminants.

The conjecture that Z(k, G; x) has an asymptotic behaviour of the above form for some constants a(k, G), b(k, G) has been stated before (see Cohen, Diaz y Diaz, Olivier [4, Conj. 1], for example). The point here is that we propose explicit expressions for a(G), b(k, G): For a finite transitive permutation group G ≠ 1 we let a(G) be the inverse of the minimal index of nontrivial elements of G. We denote by b(k, G) the number of k-rational conjugacy classes of G with minimal index. (These terms will be defined in the next section.)

Since we claim that c(k, G) > 0 for all k and G, the truth of our conjecture would in particular imply that any finite group occurs as a Galois group over any number field. At present, even this latter problem is still wide open.

Also, we will see in Corollary 2.3 that the validity of Conjecture 1.1 implies that the total number of field extensions of degree n and absolute value of the discriminant at most x should grow linearly with x. This last conjecture has been put forward by several authors, see for example Cohen [3, Conj. 9.3.5(1)].

The asymptotic behaviour of Z(k, G; x) has been the object of several investigations and speculations. Notably, Wright proved that the conjecture is true for abelian groups G (see Section 3). Cohen and his collaborators have determined the asymptotics of Z(k, G; x) for some small non-abelian groups. In collaboration with J. Klüners [8] we investigated the asymptotics of Z(k, G; x) for arbitrary nilpotent groups in regular representation, but the information obtained there is weaker than Conjecture 1.1.

The above conjecture would imply the the author’s conjecture stated in [10], where the value of a(G) was already proposed. Remark 1.2 We note that all available evidence suggests that the Conjecture 1.1 should remain true (with the same exponents a(G), b(k, G), if we restrict ourselves to counting G-extensions with prescribed loval behaviour at a finite number of places of k, provided that there exists at least one G-extension with this local behaviour. (There are examples known for which this latter condition is not satisfied.)

2. The constants a(G) and b(k, G)

We now give precise definitions of the two exponents occurring in Conjecture 1.1. The first one is purely group theoretical. For a permutation σ on a finite set Ω we define its index as

ind(σ) := |Ω| − |Ω/⟨σ⟩|,

the order of Ω minus the number of orbits of σ on Ω. Then, for a permutation group G ≠ 1 we let

(1)  a(G) := ( min{ind(σ) | 1 ≠ σ ∈ G})^{−1} ,

and we define a(1) := 0 for the trivial group. (More precisely we should have written a(G ↪ 𝔖_{n}), since a(G) also depends on the chosen permutation representation of G.)

The second constant not only depends on G but also on the number field k. Let Γ := Gal(ℚ̄/ℚ) be the absolute Galois group of ℚ and c : Γ → ℤ̂^{×} the cyclotomic character, so that δ ∈ Γ acts on the primitive roots of unity by

ζ_{n}^{δ} = ζ_{n}^{c(δ)} for ζ_{n} = exp(2πi/n) ∈ ℚ̄.

Then c factors through the commutator factor group Gal(ℚ^{ab}/ℚ), the Galois group of the maximal abelian extension of ℚ. Now let G be a finite group and σ ∈ G. We define σ^{δ} := σ^{c(δ)} for δ ∈ Γ. This defines an action of Γ on the set of elements of G (factoring through (ℤ/|G|ℤ)^{×}) which commutes with inner automorphisms (conjugation). Thus we get an induced action of Γ on the set Cl(G) of conjugacy classes of G. This action agrees with the action of Γ on the (entries of the) columns (indexed by Cl(G)) of the complex character table of G.

Assume moreover that G ≤ 𝔖_{n}. Since δ ∈ ℤ̂^{×} is invertible, the elements σ and σ^{δ} are powers of one another and thus have the same index ind(σ) = ind(σ^{δ}). So the action of Γ respects indices.

For a number field k ≤ ℚ̄, let Γ_{k} := Gal(ℚ̄/k) be its absolute Galois group. Let 1 ≠ G ≤ 𝔖_{n} be a transitive permutation group. Then we define b(k, G) to be the number of orbits of Γ_{k} on the set of those conjugacy classes of G with minimal index:

(2)  b(k, G) := |{C ∈ Cl(G) | ind(C) = a(G)^{−1}}/Γ_{k}|.

Observe that a(G) ∈ (0, 1] and b(k, G) ∈ ℕ for all k and all G ≠ 1.

Example 2.1. Let G = C_{p}, the cyclic group of prime order p, in its regular permutation representation. Then all nontrivial elements have index p − 1, which is hence the minimal index. We have b(ℚ, G) = 1, since all nontrivial elements lie in the same ℤ̂^{×}-orbit, and b(ℚ(ζ_{p}), G) = p−1, since now each element remains fixed under Gal(ℚ̄/ℚ(ζ_{p})). More generally, for a number field k denote by l the degree of k ∩ ℚ(ζ_{p}) over ℚ, that is, the degree of the maximal subfield of k contained in the field of pth roots of unity. Then the above definition shows that b(k, G) = l, which is just the same as (p − 1)/[k(ζ_{p}) : k] (see Section 3).

We determine possible values of b(k, G) for small a(G)^{−1}.

Lemma 2.2. Let G ≤ 𝔖_{n} be a transitive permutation group with a(G) = 1. Then b(k, G) = 1 for all number fields k.

Proof. First note that a(G) = 1 implies that G contains transpositions. Moreover, these are the only permutations with index 1. Thus we are done if we prove that all transpositions in G are conjugate. Let σ, τ ∈ G be two such. Since G acts transitively, we may choose a conjugate τ′ of τ whose support has (at least) one point in common with the support of σ. If τ′ ≠ σ then τ′ and σ generate a symmetric group of degree 3, but inside that all transpositions are conjugate. Hence τ and σ are conjugate. □

Denote by

Z(k, n; x) := | {K/k | (K : k) = n, N_{k/ℚ}(d_{K/k}) ≤ x} |

the total number of field extensions of k of degree n with bounded discriminant. Then we get:

Corollary 2.3. Assume Conjecture 1.1. Then for any n > 1 there exists c(k, n) > 0 with

Z(k, n; x) ∼ c(k, n) x.

Moreover, c(k, n) = ∑_{a(G)=1} c(k, G), the sum running over the transitive subgroups G of 𝔖_{n} with a(G) = 1, up to conjugation.

Proof. Indeed, Z(k, n; x) is the sum of Z(k, G; x) over all classes of transitive subgroups G of 𝔖_{n}. Now a(G) ≤ 1 for all G, and a(G) = 1 implies b(k, G) = 1 by Lemma 2.2. Since the symmetric group itself has a(𝔖_{n}) = 1 the statement follows from Conjecture 1.1. □

In particular, we conjecture that no proper subgroup G < 𝔖_{n} of 𝔖_{n} has a larger density than 𝔖_{n} itself. Note however that for any composite n there exist proper subgroups G < 𝔖_{n} with a(G) = 1, and by [10, Prop. 6.1], for any composite n not prime to 6 there exist proper subgroups G < 𝔖_{n} with Z(k, G; x) ≥ c x for an unbounded set of values x. The smallest such example is the dihedral group D_{4}, for which this statement was already shown by Baily, see [3, p. 449]. This should be compared to the Hilbert irreducibility theorem which states that the polynomials of degree n with Galois group different from 𝔖_{n} are rare.

Now let G ≤ 𝔖_{n} be a transitive permutation group with a(G) = 1/2. If G does not contain double transpositions, then b(k, G) ≤ 2. Indeed, in this case elements of minimal index are 3-cycles. Given two 3-cycles σ_{1}, σ_{2} in G, we may assume by transitivity that (up to conjugation) their supports have at least one point in common. Let H := ⟨σ_{1}, σ_{2}⟩ ≤ G and Ω the union of the supports of σ_{1}, σ_{2}. If the supports intersect in one point, then |Ω| = 5. Thus H is a transitive subgroup of 𝔖_{5} of order divisible by 3, hence equal to 𝔄_{5}, so the two elements are conjugate. If the supports intersect in at least two points, H is equal to 𝔄_{|Ω|} on Ω, so σ_{2} is conjugate to σ_{1} or its inverse in H, so in G. In any case G contains at most two classes of 3-cycles.

On the other hand, if G does contain double transpositions, there is no upper bound on b(k, G):

Lemma 2.4. For any number field k, the set

{b(k, G) | G ≤ 𝔖_{n} transitive, a(G) = 1/2, n ≥ 1}

is unbounded.

Proof. Let n = 2^{r} ≥ 4, and consider the elementary abelian 2-subgroup N of 𝔖_{n} generated by the double transpositions (2i−1, 2i)(2j−1, 2j) for 1 ≤ i < j ≤ n/2. This contains \binom{2^{r−1}}{2} double transpositions. Let H be any group of order 2^{r−1}, embedded into 𝔖_{n} via its regular permutation representation on the set of 2^{r−1} pairs (2i − 1, 2i), 1 ≤ i ≤ 2^{r−1}. The semidirect product G := ⟨N, H⟩ is an even transitive 2-group of order 2^{2^{r−1}−1} · 2^{r−1} = 2^{2^{r−1}+r−2} containing double transpositions, so a(G) = 1/2. Under the action of H, the double transpositions in N fall into at least 2^{r−2} classes. Thus b(k, G) ≥ 2^{r−2}. □

For n = 4, the construction in the proof of Lemma 2.4 gives the Klein four group, for n = 8 we get the two transitive groups 8T_{20} and 8T_{22}, with b(k, G) = 2 respectively b(k, G) = 3, and for n = 16 the construction leads to five groups, corresponding to the five groups of order 8, with 4 ≤ b(k, G) ≤ 7. It is clear that similar arguments apply for smaller values of a(G) as well.

If two transitive subgroups G_{1}, G_{2} of 𝔖_{n} have a(G_{1}) = a(G_{2}) and b(ℚ, G_{1}) = b(ℚ, G_{2}), then Conjecture 1.1 predicts that the asymptotics of their counting functions should differ by a constant. The smallest degree for which such a pair of non-isomorphic groups exists is four, with the groups G_{1} = D_{4}, G_{2} = 𝔖_{4}. The results of Cohen et al. and Bhargava cited in the next section show that

c(ℚ, 𝔖_{4})/c(ℚ, D_{4}) = 9.68838...

The next smallest cases are G_{1} = D_{5}, G_{2} = F_{20} in degree 5. For neither of these groups the asymptotics are known, but computational results seem to indicate that c(ℚ, D_{5})/c(ℚ, F_{20}) ∼ 1.5 for totally real extensions with these groups.

In degree 6, there are six groups with a(G) = 1/2, b(ℚ, G) = 1, four each with a(G) = 1/2, b(ℚ, G) = 2, respectively a(G) = 1, b(ℚ, G) = 1, and two with a(G) = 1/3, b(ℚ, G) = 1.

3. Known results

In the case of abelian groups G the asymptotic behavior of Z(k, G; x) was completely determined by Wright [11] using class field theory. He showed that in this case

Z(k, G; x) ∼ c(k, G) x^{a(G)}(log x)^{ν(k,G)−1}

for some constant c(k, G) > 0, where ν(k, G) is defined a follows. Let p denote the smallest prime dividing the order of G and n_{p} the number of elements of order p in G. Thus, n_{p} is of the form p^{r} − 1 for some r ≥ 1.

Then ν(k, G) := n_{p}/(k(ζ_{p}) : k) where ζ_{p} denotes a primitive pth root of unity. By Example 2.1 it is clear that we have ν(k, G) = b(k, G). Thus we have:

Theorem 3.1. (Wright) Conjecture 1.1 holds for abelian groups.

Example 3.2. Let G = 𝔄_{4} in its natural permutation representation of degree 4. Then a(G) = 1/2, b(k, G) = 3 if ζ_{3} := exp(2πi/3) ∈ k, b(k, G) = 2 otherwise. Cohen, Diaz y Diaz and Olivier [5, 2.7], conjecture on the basis of extensive data and heuristical arguments that

Z(k, 𝔄_{4}; x) ∼ \begin{cases}c(k, 𝔄_{4})\sqrt{x} log x & if ζ_{3} ∉ k, \\ c(k, 𝔄_{4})\sqrt{x} log^{2} x & if ζ_{3} ∈ k, \end{cases}

for constants c(k, 𝔄_{4}) > 0, in agreement with Conjecture 1.1.

By a result of Davenport and Heilbronn (see [3, p. 449]), Conjecture 1.1 holds for the symmetric group 𝔖_{3}, and by Cohen, Diaz y Diaz and Olivier [6, 2.6], it also holds for the dihedral group D_{4} of order 8 in its degree 4 permutation representation.

Bhargava [1, Thm. 5.1] has shown that the number of totally real quartic 𝔖_{4}-fields over ℚ grows asymptotically like cx, with an explicit constant c > 0.

4. A heuristic argument

Let K/ℚ be a number field of degree n. This induces a transitive permutation representation G ≤ 𝔖_{n} of the Galois group G of its Galois closure on the embeddings of K into ℚ̄. Let p be a prime dividing the discriminant d_{K/ℚ}, but not dividing |G|. Thus p is tamely ramified in K/ℚ. If p^{k} is the precise power of p dividing d_{K/ℚ}, then the inertia group at p is generated by a permutation g ∈ G with ind(g) = k (see for example [9]). Hence we have in particular a(G) ≥ 1/k. Moreover k = a(G)^{−1} does occur if the relevant inertia group is generated by a non-trivial element of G with minimal possible index. Thus if a(G) = 1/k then the possible discriminants of G-extensions of ℚ are of the form

± ∏_{i=1}^{r} p_{i}^{a_{i}} with p_{i} ∈ ℙ, a_{i} ≥ k and r ≥ 1.

In [10] we proposed a heuristic explanation for the exponent a(G) in our conjecture based on the assumption that all integers of this type are ‘equally likely’ to occur as discriminants of a G-extension of ℚ. Here we present a refinement which also takes into account the number of conjugacy classes with minimal index and then also explains the exponent b(ℚ, G).

The density of the integers of the above form is clearly dominated by the ones with a_{i} = k, 1 ≤ i ≤ r. Now assume that d = d_{K/ℚ} is an integer of this form. Then for each of the r ramified primes, the inertia group generator could lie in one of the b := b(ℚ, G) classes with minimal index. If all these cases arise (just once), then we get b^{r} extensions with discriminant d. Then the asymptotic behaviour of Z(ℚ, G; x) would be approximated by

∑_{n=1}^{x^{a(G)}} b^{ω(n)}

where ω(n) denotes the number of different prime divisors of n. The asymptotic behaviour of the latter function is well-known:

Lemma 4.1. We have

∑_{n=1}^{x^{a(G)}} b^{ω(n)} ∼ c x^{a(G)}(log x)^{b−1}

for some constant c = c(G, b) > 0.

Proof. Let’s consider the Dirichlet series

f(s) := ∑_{n≥1} \frac{b^{ω(n)}}{n^{s}} .

For s with large enough real part ℜ(s) this can be written as an Euler product over all primes p as follows:

f(s) = ∏_{p}(1 + bp^{−s} + bp^{−2s} + …)
= ∏_{p} (1 + (b − 1)p^{−s}) (1 + p^{−s} + p^{−2s} + …)
= ∏_{p} (1 + (b − 1)p^{−s}) (1 − p^{−s})^{−1} .

Now

(1 + (b − 1)p^{−s})(1 − p^{−s})^{b−1} = 1 − \binom{b}{2}p^{−2s} + … − (b − 1)(−p)^{−bs}

hence f(s) equals

∏_{p} (1 − p^{−s})^{−b} (1 − \binom{b}{2}p^{−2s} + … − (b − 1)(−p)^{−bs}) = ζ(s)^{b} g(s)

for all s with ℜ(s) > 1 where ζ is the Riemann ζ-function and g(s) is a function holomorphic in ℜ(s) ≥ 1 with g(1) ≠ 0. For example, if b = 2 then f(s) = ζ(s)^{2}/ζ(2s). Thus f(s) is holomorphic in ℜ(s) ≥ 1, except for a pole of order b at s = 1, so by a Tauberian theorem we deduce that

∑_{n=1}^{x^{a(G)}} b^{ω(n)} ∼ c′ x^{a(G)}(log x^{a(G)})^{b−1} = c x^{a(G)}(log x)^{b−1}

with a constants c, c′ > 0, as claimed. □

5. Experimental data

In this section we present the results of computational enumerations of totally real 𝔄_{5}- and 𝔖_{5}-extensions of ℚ of bounded discriminant.

5.1. The alternating group 𝔄_{5}.

We have written a program in the number theory package KANT [7] to enumerate totally real extension of ℚ of degree 5 with Galois group 𝔄_{5}. By the theorem of Hunter [3, Thm. 9.3.1], any such extension K/ℚ with discriminant d_{K/ℚ} ≤ D can be generated by a root of a (totally real) polynomial of the form

f(X) = X^{5} + a_{4}X^{4} − a_{3}X^{3} + a_{2}X^{2} + a_{1}X + a_{0},

where a_{4} ∈ {0, 1, 2} and

0 ≤ a_{3} ≤ \sqrt[4]{\frac{D}{20}} − \frac{2a_{4}^{2}}{5} .

We have used the following procedure to enumerate all such polynomials. First note that with f all of its derivatives f^{(i)}, i = 1, 2, … also have to be totally real. To obtain bounds for the coefficient a_{2} we use that the second derivative f′′ of f is totally real. Thus its discriminant, which is a quadratic polynomial in a_{2}, has to be positive. This yields bounds for a_{2}, when a_{3}, a_{4} have been chosen:

|25a_{2} + a_{4}(15a_{3} + 4a_{4}^{2})| ≤ \sqrt{2(5a_{3} + 2a_{4}^{2})^{3}}.

To bound a_{1} note that f′ has to be totally real, thus its constant coefficient a_{1} has to be such that f′ has four real zeros. This happens precisely for values of a_{1} in a (possibly empty) interval with bounds described by the values of f′ − a_{1} at zeros of f′′ (the maxima and minima of f′).

Now the discriminant of f is a polynomial d(X) ∈ ℚ[a_{1}, …, a_{4}] of degree 4 in the last coefficient a_{0}, with coefficients formal expressions in a_{1}, …, a_{4}. These coefficients can be computed explicitly at this stage already. We now make use of the desired Galois group of the polynomial: For a ∈ {0, …, m} we compute a table T stating whether the value d(a) is a square modulo m, for m ∈ {11, 63, 64, 65}. (This is inspired by the square detection algorithm described in Cohen [2, 1.7.2].) If there is no such a, then the present value of a_{1} can be discarded. Otherwise, the values a_{0} such that f is totally real lie in a finite (possibly empty) interval I. By guessing (using information from a previous value for a_{1}), or by computing a suitable zero of d(X)′ (that is, the position of the maximum of d(X)), we find one integer b in this interval I (if it is non-empty). Then all possible a_{0} can be reached successively by increasing (and decreasing) b. For each value we first check whether the discriminant can be a square by table-lookup in table T for a_{0} mod m, m ∈ {11, 63, 64, 65}. (This dramatically reduces the number of evaluations of d(X) and of integer-square-tests, and thus the running time.) If a_{0} passes these tests, we check whether d(a_{0}) is indeed the square of an integer and f is irreducible, and in the latter case output the polynomial f.

The polynomials obtained this way are then filtered according to their Galois group (which at this point can be one of 𝔄_{5}, D_{5} or C_{5}). Then the 𝔄_{5}-fields are tested for isomorphism. (To speed up this step, it is sensible to first find a reduced generating polynomial for each 𝔄_{5}- field, using the polredabs command in Pari, for example, and discard doubles.) Note that fields with the solvable groups D_{5} and C_{5} can be enumerated much more efficiently using methods from class field theory, see for example [3, Section 9], or [4].

We give the results of these computations in the form of a diagram in doubly logarithmic scale, showing numbers of field extensions with group 𝔄_{5}. We also print a line with slope a(𝔄_{5}) = 1/2. Since b(ℚ, 𝔄_{5}) = 2, our conjecture predicts that the number of extensions should grow proportionally to \sqrt{x} log x.

Table 1 gives the distribution of all totally real 𝔄_{5}-fields of discriminant d_{K} ≤ 2^{37} = 1.37..10^{11}.

Table 1. First 9 933 totally real 𝔄_{5}-extensions

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . . . . .  .  .  .  .  . 6 7 8 9 10 11 log D .... 1234 log Z  \begin{matrix}D & Z(ℚ, 𝔄_{5}; D) \\ 2^{22} & 1 \\ 2^{23} & 7 \\ 2^{24} & 16 \\ 2^{25} & 30 \\ 2^{26} & 57 \\ 2^{27} & 104 \\ 2^{28} & 159 \\ 2^{29} & 251 \\ 2^{30} & 421 \\ 2^{31} & 681 \\ 2^{32} & 1095 \\ 2^{33} & 1685 \\ 2^{34} & 2612 \\ 2^{35} & 4094 \\ 2^{36} & 6371 \\ 2^{37} & 9933 \end{matrix}

5.2. The symmetric group 𝔖_{5}.

For the symmetric group 𝔖_{5}, a KANT-program was used to compute all totally real degree 5 extensions of ℚ with symmetric Galois group up to discriminant 10^{9}. The results are displayed in Table 2. Here Conjecture 1.1 predicts a linear growth of Z(ℚ, 𝔖_{5}; x) with x.

Table 2. First 2 537 415 totally real 𝔖_{5}-extensions

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . . . . .  .  .  .  . 4 5 6 7 8 9 log D ...... 123456 log Z  \begin{matrix}D & Z(ℚ, 𝔖_{5}; D) \\ 2^{22} & 3 035 \\ 2^{23} & 7 488 \\ 2^{24} & 18 211 \\ 2^{25} & 43 112 \\ 2^{26} & 100 077 \\ 2^{27} & 229 657 \\ 2^{28} & 518 546 \\ 2^{29} & 1 153 555 \\ 2^{30} & 2 537 415 \end{matrix}

In [10, §7] we gave a heuristical explanation of the exponent a(G) in the conjecture. For that, we postulated that the set of integers which occur as discriminants of field extensions with a given group G with a(G) = 1 has positive density among all integers. In particular, one might expect that the multiplicative structure of discriminants (in a range) resembles that of all integers (in a range).

We used the data of the first 10^{6} totally real 𝔖_{5}-extensions to test this assumption. More precisely, we checked the divisibility of the discriminants by the first 100 primes. A proportion of 1/p of all integers (in a range) is divisible by a prime p. It turns out that in our range for all relatively small primes except 2, the actual proportion is less than the expected one, slightly increasing with p. But the deviation is at most by 17% (for the prime p = 17), and quite a bit smaller for most other primes.

5.3. Regular groups in degree 8.

There are two non-abelian regular permutation groups of degree 8: the dihedral and the quaternion group of order 8. Both of them have a(G) = 1/4, and elements with minimal index are precisely the involutions. But while the quaternion group Q_{8} has just one involution, the central one, the dihedral group D_{4} has three conjugacy classes of involutions. Thus b(k, Q_{8}) = 1, b(k, D_{4}) = 3 for any number field k. Computations in KANT by J. Klüners indicate that indeed the number of Q_{8}-extensions grows linearly with x^{1/4}, while the number of D_{4}-extensions of degree 8 grows proportionally to x^{1/4}(log x)^{2}, as would be expected by Conjecture 1.1.

Acknowledgement: I like to thank Jürgen Klüners for computing large tables of number fields of degree 8 with groups Q_{8} and D_{4}.

References

[1] M. Bhargava, Higher Composition Laws, Dissertation, Princeton University, Princeton, 2001.

[2] H. Cohen, A Course in Computational Algebraic Number Theory, Springer, Berlin – Heidelberg – New York, 1993.

[3] H. Cohen, Advanced Topics in Computational Number Theory, Springer, Berlin – Heidelberg – New York, 2000.

[4] H. Cohen, F. Diaz y Diaz, M. Olivier, Counting discriminants of number fields, MSRI preprint 2000-026, 2000.

[5] H. Cohen, F. Diaz y Diaz, M. Olivier, A survey of discriminant counting, in: Algorithmic Number Theory, C. Fieker, D.R. Kohel, eds., LNCS 2369, Springer, Berlin, 2002, pp. 80–94.

[6] H. Cohen, F. Diaz y Diaz, M. Olivier, Enumerating quartic dihedral extensions, Compositio Math. 133 (2002), 65–93.

[7] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, K. Wildanger, KANT V4, J. Symb. Comput. 24 (1997), 267– 283.

[8] J. Klüners, G. Malle, Counting nilpotent Galois extensions, submitted, 2001.

[9] H. Koch, Number Theory - Algebraic Numbers and Functions, Amer. Math. Soc., Providence, Rhode Island, 2000.

[10] G. Malle, On the distribution of Galois groups, J. Number Theory 92 (2002), 315–329.

[11] D. Wright, Distribution of discriminants of abelian extensions, Proc. London Math. Soc. 58 (1989), 17–50.

FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany.

E-mail address: malle@mathematik.uni-kassel.de