ON THE DISTRIBUTION OF GALOIS GROUPS

Gunter Malle

Universität Kassel

We propose a conjecture on the distribution of number fields with given Galois group and bounded norm of the discriminant. This conjecture is known to hold for abelian groups. We give some evidence relating the general case to the composition formula for discriminants, give a heuristic argument in favor of the conjecture and present some computational data.

1. Introduction

In this paper we are interested in the distribution of number fields with a given Galois group. We are led to formulate a conjecture on the density of fields having given discriminant and Galois group. More precisely, 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(L/K) = G if L/K is a field extension such the Galois group of the Galois closure L̂/K viewed as permutation group on the set of embeddings of L into L̂ is permutation isomorphic to G. We let

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

be the number of field extensions of K (inside a fixed algebraic closure ℚ̄) of degree n with Galois group permutation isomorphic to G (as explained above) and absolute value of the norm of the discriminant d(L/K) bounded above by x. It is well known that the number of extensions of K with bounded norm of the discriminant is finite, hence Z(K, G; x) is finite for all G, K and x ≥ 1.

In order to be able to state our conjecture we have to introduce a group theoretic invariant: For a permutation g (on n points) we define its index as

ind(g) := n − the number of orbits of g on {1, …, n} .

Note that the index is not changed when g is considered as an element of a bigger symmetric group in the natural way. For example, ind(g) = 1 for transpositions g ∈ 𝔖_{n} for any n. Then, for a permutation group G ≠ 1 we let

a(G) := ( min{ind(g) | g ∈ G^{#}})^{−1}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 11R32, 11R29, 12-04.

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

where G^{#} = 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.) Thus for any (transitive permutation) subgroup H ≤ G we have a(H) ≤ a(G).

We make 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. Let G be a transitive permutation group and K a number field. Then for all ϵ > 0 there exist constants c_{1}(K, G), c_{2}(K, G, ϵ) ∈ ℝ_{>0} such that

c_{1}(K, G) x^{a(G)} ≤ Z(K, G; x) < c_{2}(K, G, ϵ) x^{a(G)+ϵ}

for all large enough x.

Note that this conjecture would imply in particular that any finite group occurs as a Galois group over any number field, i.e., a positive solution to the inverse problem of Galois theory for number fields. (Thus we have little hope of seeing a proof for it in the near future.) In particular the total number of field extensions of degree n and absolute value of the discriminant at most x should grow at least like x and less than x^{1+ϵ} for all ϵ > 0. This last conjecture has been proposed by several authors, see for example [3], Conjecture 9.3.5(1).

The non-trivial permutations with smallest possible index are the transpositions. Now by a theorem of Jordan a primitive transitive permutation group containing a transposition coincides with 𝔖_{n}. Thus one consequence of the validity of the conjecture would be that among primitive extensions of degree n of bounded discriminant, those with Galois group the symmetric group 𝔖_{n} have density 1. For example, all groups of prime degree are primitive, thus for prime degree n = p asymptotically almost all extensions should have Galois group 𝔖_{p}.

On the other hand, our conjecture predicts that for any composite degree n there exist groups G ≠ 𝔖_{n} such that Z(K, G; x) grows at least like x. We will in fact prove this statement when n > 3 is not prime to 6 in Proposition 6.1. The smallest such example is the dihedral group D_{4}, for which this statement was already shown by Baily [1], 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.

Moreover we have ind(g) ≤ n − 1 for g ∈ 𝔖_{n}, so a(G) ≥ (n − 1)^{−1} for G ≤ 𝔖_{n}. Hence for any transitive group G on n letters the number of extensions with group G should grow at least like c \sqrt[n−1]{x} with the discriminant.

In the remainder of the paper we will collect the evidence for the validity of our conjecture. This consists of cases in which the conjecture is known to hold, of certain compatibility results for suitable composite groups, a heuristic argument, and finally of some computational evidence.

Acknowledgement: I like to thank Jürgen Klüners for proposing the formulation of the stronger form 2.1 of the conjecture, and Claus Fieker and him for computing large tables of totally real number fields with groups 𝔄_{4} and D_{5}, see Section 8.

2. A stronger version of the conjecture

We present a stronger form of our main conjecture which seems more suitable for inductive purposes. Recall that, throughout the paper, if G is a transitive permutation group then we say that L/K has group G if the Galois group of the Galois closure of L/K is isomorphic to G and L is the fixed field of (the image of) a point-stabilizer in G. For a number field K let ℙ(K) denote the set of prime divisors including the infinite ones. Then for any subset S ⊂ ℙ(K) we define

Z(K, S, G; x) := |{L/K | Gal(L/K) = G, |N_{K/ℚ}(d(L/K))| ≤ x,

L/K not ramified in S}|

to be the number of field extensions of K unramified in S of degree n with Galois group isomorphic to G and norm of the discriminant bounded above by x. Then we may state the following sharpening of the conjecture from the introduction (which is obtained as the case S = ∅):

Conjecture 2.1. Let G be a transitive permutation group, K a number field and S ⊂ ℙ(K) finite. Then for all ϵ > 0 there exist constants c_{1}(K, S, G), c_{2}(K, G, ϵ) ∈ ℝ_{>0} such that

c_{1}(K, S, G) x^{a(G)} ≤ Z(K, S, G; x) < c_{2}(K, G, ϵ) x^{a(G)+ϵ}

for all large enough x.

If the conjecture holds, then clearly by increasing c_{2}(K, G, ϵ) we may assume that the upper bound holds for all x > 0.

3. The case of abelian extensions

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. That paper also contains references to previous work and related results.

To formulate the statement in this case, we need to introduce a bit of further notation. Let G ≠ 1 be an abelian group and p be the smallest prime dividing the order of G. We denote by n_{p} the number of elements of order p in G. Thus, n_{p} is of the form p^{k} − 1 for some k ≥ 1. We set ν(K, G) := n_{p}/(K(ζ_{p}) : K) where ζ_{p} denotes a primitive pth root of unity. Thus ν(K, G) is a positive integer. Also we write α(G) := |G|(1 − p^{−1}).

Theorem (Wright [11]). Let G be a finite abelian group, K a number field. Then there exists a constant c̃(K, G) > 0 such that

Z(K, G; x) ∼ c̃(K, G) x^{1/α(G)}(log x)^{ν(K,G)−1}.

Here we write f ∼ g for functions f, g : ℝ → ℝ_{>0} if lim_{x→∞} f(x)/g(x) = 1.

Corollary. The conjecture is true for abelian groups.

Proof. Indeed, the only faithful transitive permutation representation of a finite abelian group G is the regular one of degree |G|. The index of any element g ∈ G of order m in this representation equals

|G| − number of orbits of g = |G| − \frac{|G|}{m} = \frac{m − 1}{m}|G| .

Its minimum for nontrivial elements g ∈ G is attained for minimal m ≠ 1, that is, for m = p where p is the smallest prime divisor of |G|. Thus a(G) = 1/α(G) as claimed. □

It follows from the arguments in [11], Sect. 7, that even the stronger Conjecture 2.1 holds for abelian extensions. By the remarks on p. 47 of loc. cit. it is sufficient for this that there exists at least one extension of K with group G not ramified in S, but this is clear for any abelian G and any finite S.

4. Inductive consistency with direct products

In the next two sections we want to compare the statement of Conjecture 2.1 for different groups. In essence we verify that the conjecture in a certain sense is consistent with the composition formula for discriminants.

We first compare our prediction for finite groups G_{1}, G_{2} and for their direct product G_{1}×G_{2}. So let G_{1}, G_{2} be transitive permutation groups on n_{1}, n_{2} points respectively. The direct product G_{1} ×G_{2} acts naturally on n_{1}n_{2} points. Then with respect to these permutation representations we get:

Lemma 4.1. We have a(G_{1} × G_{2}) = max(\frac{a(G_{1})}{n_{2}}, \frac{a(G_{2})}{n_{1}}).

Proof. Consider a permutation g = (g_{1}, g_{2}) ∈ G_{1} × G_{2}. Note that g acts on the n_{1}n_{2} points by (g_{1}, g_{2})(u_{1}, u_{2}) = (g_{1}u_{1}, g_{2}u_{2}). Hence the number of orbits of g is not smaller than the number of orbits of (g_{1}, 1) or of (1, g_{2}). But the latter two elements have index ind(g_{1}, 1) = n_{2}ind(g_{1}), respectively ind(1, g_{2}) = n_{1}ind(g_{2}). Thus the assertion follows. □

The problem for deriving a lower bound for G_{1}×G_{2}-extensions lies in the possibility that all (or most) G_{1}- and G_{2}-extensions of K might contain a common subfield unramified over K, and thus not enough extensions with the direct product exist. Although this seems unlikely to happen, it has to be excluded by adding suitable hypotheses.

Proposition 4.2. Assume that K has no nontrivial unramified Galois extension with Galois group a common quotient of G_{1} and G_{2}. If n_{1}a(G_{1}) ≤ n_{2}a(G_{2}) and the lower bound of Conjecture 2.1 holds for G_{2} then the lower bound from Conjecture 2.1 holds for G := G_{1} × G_{2}, that is, for all finite S ⊂ ℙ(K) there exists c(K, S, G) > 0 with

c(K, S, G) x^{a(G)} ≤ Z(K, S, G; x)

for all large enough x.

Proof. By assumption and Lemma 4.1 we have that a(G_{1} × G_{2}) = a(G_{2})/n_{1}.

On the side of extensions, composition defines a natural surjective map from the set of pairs of linearly disjoint G_{1}, G_{2}-extensions of K to G_{1} ×G_{2}-extensions of K (all considered inside a fixed algebraic closure of K), with fibers of fixed finite size.

Fix a finite S ⊂ ℙ(K). By assumption there exists a G_{1}-extension L_{1}/K ramified in S_{1} ⊂ ℙ(K), such that S ∩ S_{1} = ∅. Let L_{2} be any extension of K with group G_{2}, unramified in S_{2} := S ∪ S_{1}. Then L_{1}/K and L_{2}/K are ramified in disjoint sets. So, since K has no proper unramified extension with group a common quotient of G_{1} and G_{2}, they have linearly disjoint Galois closures. Thus L_{1}L_{2}/K has group G_{1} × G_{2} and discriminant

N_{K/ℚ}(d(L_{1}L_{2}/K)) = N_{K/ℚ}(d(L_{2}/K))^{n_{1}}N_{L_{2}/ℚ}(d(L_{1}L_{2}/L_{2}))

by the composition formula for discriminants. Note that the second factor is bounded above by (in fact equal to) d := N_{K/ℚ}(d(L_{1}/K))^{n_{2}} independently of L_{2}.

By assumption there exists a c > 0 such that Z(K, S_{2}, G_{2}; x) ≥ cx^{a(G_{2})} = cx^{n_{1}a(G)} for large enough x. Clearly we obtain a lower bound for the number of G_{1} × G_{2}- extensions of K by just counting those containing L_{1}. The above discriminant formula implies that the number of such G_{1} × G_{2}-extensions unramified in S satisfies

Z(K, S, G; x) ≥ Z(K, S_{2}, G_{2}; (x/d)^{1/n_{1}}) ≥ c (x/d)^{a(G)} = c′x^{a(G)}

with c′ := cd^{−a(G)} > 0, as claimed. □

The proof remains valid if the assumption on unramified extensions of K is replaced by the assumption that G_{1}^{r} occurs as Galois group over K, for suitably large r (since all we need is a G_{1}-extension of K no subextension of which is unramified).

Obviously our proof gives the lower estimate in the theorem of Wright for fields K with class number 1, since the case of cyclic groups of prime power order is easily seen to hold.

It seems that in order to obtain the correct upper bound we would have to have better control on unramified extensions of the various fields L_{1} which can occur inside L_{1}L_{2}/L_{1}. But even without this we can prove a weaker result (without assumption on K):

Proposition 4.3. Assume that the upper bound from Conjecture 2.1 holds for G_{1} and G_{2}. Let G = G_{1} × G_{2}. Then for ϵ > 0 there exists c(K, G, ϵ) > 0 with

Z(K, S, G; x) < c(K, G, ϵ) x^{\frac{a(G_{1})}{n_{2}} + \frac{a(G_{2})}{n_{1}} +ϵ}

for all large enough x.

Proof. Indeed, if L_{1}, L_{2} are extensions of K then by the composition formula

N_{K/ℚ}(d(L_{1}L_{2}/K)) ≥ max(N_{K/ℚ}(d(L_{2}/K))^{n_{1}}, N_{K/ℚ}(d(L_{1}/K))^{n_{2}}) .

If we assume that there are at most c_{i} x^{a(G_{i})+ϵ} extensions of K with group G_{i}, the number of pairs of extensions with groups (G_{1}, G_{2}) and thus the number of extensions

with the direct product grows at most like x^{\frac{a(G_{1})}{n_{2}} + \frac{a(G_{2})}{n_{1}} +ϵ}. □

5. Inductive consistency with wreath products

Let G ≤ 𝔖_{n}, H ≤ 𝔖_{k} be transitive permutation groups on n respectively k ≥ 2 letters. The wreath product G̃ = H ≀ G has a transitive permutation representation of degree nk, imprimitive on n blocks of k letters. It contains a subgroup which is permutation isomorphic to H (on the first k of the nk letters).

Lemma 5.1. We have a(H ≀ G) = a(H).

Proof. Let g = (h, g_{1}) ∈ H ≀ G where h = (h_{1}, …, h_{n}) ∈ H^{n} and g_{1} is the image of g under the projection to the complement G. If g_{1} ≠ 1 then g exchanges at least two blocks, so the number of orbits is at most (n − 2)k + k = (n − 1)k. On the other hand, if g_{1} = 1, h_{2} = … = h_{n} = 1 then g has at least (n − 1)k + 1 orbits. Thus we may assume g_{1} = 1, and then clearly the minimal index is attained in the case h_{2} = … = h_{n} = 1, whence a(H ≀ G) = a(H). □

Under a suitable boundedness condition on the constants appearing in the upper estimate of the conjecture, we get consistency with wreath products:

Proposition 5.2. Assume that there exist e_{G}, c(K, G) ≥ 0 such that Z(K, G; x) ≤ c(K, G)x^{e_{G}}. Furthermore, assume there exist e_{H}, c(H) and δ_{0} < ke_{H} − e_{G} such that for all L/K with group G we have Z(L, H; x) ≤ c(H)d_{L}^{δ_{0}}x^{e_{H}}, where d_{L} := |N_{K/ℚ}(d(L/K))|. Then

Z(K, G̃; x) < c(K, G̃) x^{e_{H}} for all x > 0.

Proof. If N/K is an extension of degree nk with group G̃ then there exists an intermediate extension L/K of degree n with group G such that N/L of degree k has group H. By the composition formula the discriminant of N/K is given by d(N/K) = d(L/K)^{k}N_{L/K}(d(N/L)), hence

N_{K/ℚ}(d(N/K)) = N_{K/ℚ}(d(L/K))^{k}N_{L/ℚ}(d(N/L)) .

Write L(x) for the set of field extensions L/K (inside a fixed algebraic closure of K) with group G and d_{L}^{k} ≤ x. Then our previous discussion shows

\begin{aligned}Z(K, G̃; x) & ≤ ∑_{L∈L(x)} Z(L, H; x/d_{L}^{k}) \\ & ≤ ∑_{L∈L(x)} c(H) d_{L}^{δ_{0}} (\frac{x}{d_{L}^{k}} )^{e_{H}} = c(H) x^{e_{H}} ∑_{L∈L(x)} d_{L}^{δ_{0}−ke_{H}} . \end{aligned}

Define l(d) := |{L ∈ L(x) | d_{L} = d}| and y := ⌊x^{1/k}⌋. Then clearly

(*)  ∑_{L∈L(x)} d_{L}^{δ_{0}−ke_{H}} = ∑_{d=1}^{y} \frac{l(d)}{d^{ke_{H}−δ_{0}}} .

But by the assumption for G we have

∑_{d=1}^{y} l(d) = Z(K, G; y) ≤ c(K, G) y^{e_{G}} < c(K, G) y^{ke_{H}−δ_{0}} .

Under this condition an easy argument shows that the Dirichlet series corresponding to the right hand side of (*) converges (see for example Lemma 57.B in [10]). Thus we finally obtain

Z(K, G̃; x) < c(K, G̃) x^{e_{H}}

for a suitable constant c(K, G̃) > 0. □

We can apply this for example in the case that the conjecture is known to hold for H and some information is available for G:

Corollary 5.3. Assume that the upper bound in Conjecture 2.1 holds for H and there exists δ_{0} > 0 such that for any fixed ϵ > 0 we have c_{2}(L, H, ϵ) ≤ c(H, ϵ) d_{L}^{δ_{0}} for all L/K with group G, where d_{L} := |N_{K/ℚ}(d(L/K))|. Assume moreover that there exists e_{G} < ka(H) − δ_{0} with Z(K, G; x) ≤ c(K, G; ϵ) x^{e_{G}+ϵ}. Then the upper bound from the conjecture holds for G̃ = H ≀ G, that is, for all ϵ > 0 there exists c(K, G̃, ϵ) > 0 with

Z(K, G̃; x) < c(K, G̃, ϵ) x^{a(G̃)+ϵ} for all x > 0.

Proof. For ϵ > 0 we may choose ϵ′ > 0 such that e_{G} +ϵ′ < ka(H)+kϵ−δ_{0}. Now apply Proposition 5.2 with e_{H} := a(H) + ϵ and use a(G̃) = a(H) from Lemma 5.1. □

For example, when the conjecture holds for G then we may choose e_{G} = a(G) and then for small enough δ_{0} (δ_{0} < \frac{1}{k−1} will always do) we have

e_{G} = a(G) ≤ 1 < ka(H) − δ_{0}

as required in the Corollary, since a(H) ≥ \frac{1}{k−1} for all H.

Note that the Corollary remains valid with G̃ replaced by any subgroup U ≤ H ≀ G which surjects onto G with kernel containing a copy of H, with the same proof.

In the situation of wreath products it is not clear how to deduce the lower bound for Z(K, S, G̃; x); in fact, it not even clear that there has to exist one single extension with group G̃.

Nevertheless, assuming the lower bound holds for G and H, we can make the following observations. By assumption there exists a Galois extension L̂/K with group G. Denote by L/K a stem field of degree n corresponding to the chosen permutation representation of G. Then for any extension N/L of degree k with Galois group (of the Galois closure N̂/L) equal to H the Galois group (of the Galois closure) of N/K is a subgroup of G̃. Clearly, there is a fixed finite number r, only depending on G and H, such that at most r different extensions N/L lead to isomorphic extensions N/K.

The first factor in the composition formula for the discriminant of N/K only depends on the fixed G-extension L, but not on N. By our conjecture the number of fields N/L with N_{L/ℚ}(d(N/L)) ≤ x grows at least like c x^{a(H)}. Hence we obtain at least c′ x^{a(H)} fields whose Galois group is a subgroup of G̃, which by Lemma 5.1 equals c′ x^{a(G̃)}.

In practice the overwhelming majority of composite extension N/K as above actually turn out to have the full wreath product as Galois group. If we knew that at least a positive fraction of extensions N/L, for some L/K, has this property, then the lower bound for Z(K, S, G̃; x) would follow from these considerations.

6. Some unconditional bounds

The argument in the previous section shows at least the following, without any recourse to our conjecture:

Proposition 6.1. Let K be a number field and n = 2k > 2 or n = 3k > 3. Then the number of field extensions of K of degree n with Galois group not the symmetric group grows at least linearly with the absolute value of the discriminant. In particular there exists a transitive subgroup G < 𝔖_{n} with Z(K, G; x) ≥ c x for an unbounded set of values x.

Proof. By the Hilbert irreducibility theorem there certainly exists an extension L/K of degree k with group 𝔖_{k}. Clearly the number of Z_{2}-extensions of L grows linearly with the discriminant, and by the Theorem of Davenport and Heilbronn [6] the number of 𝔖_{3}-extensions also grows linearly. But a(Z_{2}) = a(𝔖_{3}) = 1. Hence we may apply the above argument to the wreath product of 𝔖_{k} with Z_{2} = 𝔖_{2}, resp. with 𝔖_{3} to show that the number of extensions of K of degree n and with group contained in 𝔖_{n/k} ≀𝔖_{k} grows at least linearly. Since there are only finitely many subgroups of 𝔖_{n/k} ≀ 𝔖_{k}, for at least one of them, say G, we must have Z(K, G; x) ≥ c x for arbitrarily large x. □

If we can realize the 𝔖_{k}-extension L/K unramified in a given finite subset S ⊂ ℙ(K), then we even obtain Z(K, S, G; x) ≥ c x, since there exist sufficiently many quadratic extensions of L unramified in any given finite set of primes.

An upper bound on the number of Galois extensions of ℚ of fixed degree n can be obtained from Hunter’s theorem, counting polynomials with bounded coefficients (see [3, Prop. 9.3.4]):

Proposition (Cohen). The number of non-isomorphic number fields of degree n and discriminant bounded in absolute value by x is at most equal to c_{n}x^{(n+2)/4} for some constant c_{n} > 0. In particular, for any transitive group G ≤ 𝔖_{n} we have Z(ℚ, G; x) ≤ c_{n}x^{(n+2)/4}.

For symmetric groups of arbitrary degree, or more generally for groups G for which a Galois realization with the rigidity method has been obtained, we can deduce weak lower bounds on Z(ℚ, G; x) as follows. Let’s explain this on the example of 𝔖_{n}:

Proposition 6.2. For all n ≥ 2 there exists a constant c_{n} > 0 such that

Z(ℚ, 𝔖_{n}; x) ≥ c_{n}x^{1/n}

for all sufficiently large x.

Proof. By [9, Thm. I.9.4] there exists an 𝔖_{n}-extension of ℚ(t) with generating polynomial f(t, X) := X^{n} − t(nX − n + 1) ∈ ℤ[t, X]. The polynomial f has discriminant

(−1)^{\binom{n}{2}}(n − 1)^{n−1}n^{n}t^{n−1}(1 − t).

If n is odd, then for any square-free number u prime to n(n − 1) the discriminant of f(1 − u, X) has square-free part ±u times the square-free part of n, hence all such specializations define distinct extensions of ℚ. By imposing suitable congruence conditions on u we can moreover guarantee that f(1 − u, X) has Galois group 𝔖_{n} over ℚ. Since the square-free numbers have positive density in the set of all integers we arrive at the desired conclusion.

A similar argument shows the assertion for even n. □

7. A heuristic approach

There is a well-known connection between the prime powers occurring in the discriminant of a number field and the indices of elements of the Galois group, which we briefly review. Let L/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 conjugates of L. Let p be a prime dividing the discriminant d(L/ℚ), but not dividing |G|. Thus L/ℚ is tamely ramified in p. If p^{k} is the precise power of p dividing d(L/ℚ), then the inertia group at p is generated by a permutation g ∈ G with ind(g) = k (see for example [8]). 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 are of the form ∏_{i=1}^{r} p_{i}^{a_{i}} with p_{i} ∈ ℙ, a_{i} ≥ k and r ≥ 1. Such integers are called k-powerful and have been intensively investigated. It is known by [7] that their number grows like γ_{k}x^{1/k} plus lower order terms, for some γ_{k} > 0. For fixed G let us write n(d) for the number of fields with group G and absolute value of the discriminant equal to d. If n(d) is bounded above, but non-zero for some subset of positive density, then we obtain that Z(ℚ, G; x) grows like x^{a(G)}, as asserted in the conjecture. In fact, it is sufficient to assume that max{n(d)|d ≤ x} grows slowly enough with x to still obtain the statement from our conjecture.

8. Small groups and experimental data

All of our examples are over K = ℚ. They concern some small nonabelian groups for which theoretical results have been obtained, as well as some computational evidence.

8A. Groups of degree at most 4.

The non-abelian permutation groups of smallest degree are 𝔖_{3}, D_{4}, 𝔄_{4} and 𝔖_{4}. For 𝔖_{3} it follows from the work of Davenport and Heilbronn [6] that

Z(ℚ, 𝔖_{3}; x) ∼ \frac{1}{ζ(3)} x ,

consistent with our conjecture. The groups of degree 4 were first investigated by Baily [1] who obtained partial results. After submission of this paper the author learned of complete results which were obtained by Cohen, Diaz y Diaz and Olivier. By the result of Wright we only have to consider the three non-abelian groups, where the following asymptotic results are now known (see the announcement in [4]):

\begin{matrix}Z(ℚ, D_{4}; x) ∼ & c(D_{4}) x, \\ Z(ℚ, 𝔄_{4}; x) ∼ & c(𝔄_{4})\sqrt{x} log(x), \\ Z(ℚ, 𝔖_{4}; x) ∼ & c(𝔖_{4})x. \end{matrix}

Since D_{4} and 𝔖_{4} contain transpositions, while 𝔄_{4} doesn’t, we have a(G) = 1 in the first case, and a(G) = 1/2 in the second. Hence the results of Baily and Cohen et al. are in complete agreement with the conjecture.

8B. Experimental evidence in degree 5.

In the following sections, we give some experimental data in the form of diagrams, in doubly logarithmic scale, giving numbers of field extensions with given group G. In each diagram we also print a line with slope a(G); by our conjecture the number of extensions should grow with this rate.

The non-abelian transitive permutation groups on five letters are 𝔖_{5}, 𝔄_{5}, the Frobenius group F_{20} of order 20 and the dihedral group D_{5} on five letters, with a(𝔖_{5}) = 1, a(𝔄_{5}) = a(F_{20}) = a(D_{5}) = 1/2. For the case D_{5} Jürgen Klüners has computed all totally real extensions of ℚ of discriminant at most 10^{12}. The data are shown in Table 8.1 and indicate indeed a growth of Z(ℚ, D_{5}; x) like \sqrt{x}.

Table 8.1. First 7 312 non-real and 13 108 totally real D_{5}-extensions.

._{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . . . . . . 4  6  8  10 log D ..... 1 2 3 4 log Z  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . . . . . . . . . 6  8  10 12 log D ..... 1 2 3 4 log Z

Table 8.2. First 1 090 totally real 𝔄_{5}-extensions.

._{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . .  .  .  .  . 6  7  8  9  10 log D .... 1 2 3 4 log Z

Finally, for the symmetric and alternating group on five letters we have written a KANT-program [5] to enumerate all totally real extensions up to discriminant 2^{32} in case of 𝔄_{5} and up to 2^{27} in case of 𝔖_{5}. For 𝔖_{5}-extensions with one real embedding we have extracted the data in Table 8.3 from the list of field extensions of degree 5 in [12]. Again the Tables 8.2 and 8.3 do not seem to contradict the conjecture, although it would seem desirable to collect some more data for a more convincing picture.

Table 8.3. First 152 373 complex and 229 483 totally real 𝔖_{5}-extensions.

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . .  .  .  .  . 3  4  5  6  7 log D ..... 1 2 3 4 5 log Z  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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . .  .  .  .  . 4  5  6  7  8 log D ..... 1 2 3 4 5 log Z

8C. Experimental evidence in degree 6.

The regular representation of the symmetric group 𝔖_{3} has degree 6. Let L/ℚ be a number field of degree 3 with group 𝔖_{3} and N/ℚ its Galois closure of degree 6. By comparing the behavior of ramified primes it is easy to verify that

d(N/ℚ) = d(L/ℚ)^{2}d(K/ℚ),

where K/ℚ denotes the quadratic subfield of N/ℚ. In particular |d(N/ℚ)| ≥ d(L/ℚ)^{2}. This allows to obtain a large number of degree 6 extensions with group 𝔖_{3} by starting from degree 3 extensions. Using Belabas’ algorithm described in [3, 8.7] it is easy to enumerate all totally real 𝔖_{3}-extensions of ℚ of degree 3 with d(L/ℚ) ≤ 10^{10}, say (there are 661 432 230 such extensions). This in turn gives all 492 335 totally real 𝔖_{3}- extensions N/ℚ of degree 6 with d(N/ℚ) ≤ 10^{20}. The results are given in Table 8.4 and seem to be in accordance with the conjecture, since a(𝔖_{3}) = 1/3 in its transitive degree 6 representation.

Table 8.4. First 492 335 totally real 𝔖_{3}-extensions of degree 6

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . . . .  .  .  .  .  .  . 6  8  10 12 14 16 18 20 log D ..... 1 2 3 4 5 log Z

Similarly we determined the 1 353 065 totally real 𝔖_{3} × 2-extensions of degree 6 with smallest discriminant. The result is given in Table 8.5. Here a(𝔖_{3} × 2) = 1.

Table 8.5. First 1 353 065 totally real 𝔖_{3} × 2-extensions

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . .  .  .  .  .  . 8  10 12 14 16 log D ...... 1 2 3 4 5 6 log Z

8D. Experimental evidence in degree 7.

It does not seem within reach of present computer power to enumerate a sizeable number of field extensions of degree 7 with non-solvable Galois group. Nevertheless we could use a KANT-program written by Jürgen Klüners to determine the smallest 6083 totally real D_{7}-extensions, using class field theory. Here our conjecture predicts a growth like \sqrt[3]{x}, and our experimental results in Table 8.6 seem in accordance with this expectation.

Table 8.6. First 6 083 totally real D_{7}-extensions.

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} . . . . . . . . . . . . . . . . . . . . . . .  .  .  .  .  .  . 11 12 13 14 15 16 17 log D .... 1 2 3 4 log Z

References

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

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