Computation of 2-groups of narrow logarithmic divisor classes of number

fields ^{∗}

Jean-François Jaulent, Sebastian Pauli, Michael E. Pohst & Florence Soriano–Gafiuk,

Abstract. We present an algorithm for computing the 2-group C̃ℓ_{F}^{res} of narrow logarithmic divisor classes of degree 0 for number fields F. As an application, we compute in some cases the 2-rank of the wild kernel WK_{2}(F) and the 2-rank of its subgroup K_{2}^{∞}(F) := ∩_{n≥1}K_{2}^{n}(F) of infinite height elements in K_{2}(F).

Résumé. Nous présentons un algorithme de calcul du 2-groupe des classes logarithmiques de degréé nul au sens restreint C̃ℓ_{F}^{res} pour tout corps de nombres F. Nous en déduisons sous certaines hypothèses les 2-rangs du noyau sauvage WK_{2}(F) ainsi que du sous-groupe K_{2}^{∞}(F) := ∩_{n≥1}K_{2}^{n}(F) des éléments de hauteur infinie dans K_{2}(F).

1 Introduction

In [8] J.-F. Jaulent pointed out that the wild kernel WK_{2}(F) of a number field F can also be studied via logarithmic class groups, the arithmetic of which he therefore developed in [9].

More precisely, if F contains a primitive 2ℓ-th root of unity, the ℓ-rank of WK_{2}(F) coincides with the ℓ-rank of the logarithmic class group C̃ℓ_{F}. So an algorithm for computing C̃ℓ_{F} for Galois extensions F was developed first in [4] and later generalized and improved for arbitrary number field in [3].

In case the prime ℓ is odd and the field F does not contain a primitive ℓth root of unity one considers the cyclotomic extension F′ := F (ζ_{ℓ}), uses the isomorphism

μ_{ℓ} ⊗ C̃ℓ_{F′} ≃ WK_{2}(F′)/WK_{2}(F′)^{ℓ} ,

and gets back to F via the so-called transfer (see [11] and [16]). The algorithmic aspect of this is treated in [14].

In case ℓ = 2, whenever the condition ζ_{2ℓ} ∈ F is not fulfilled, the relationship between logarithmic classes and exotic symbols is more intricate. For instance, when the number field F has a real embedding, F. Soriano-Gafiuk observed in [15] that one may then define a narrow version of the logarithmic class group by analogy with the classical ideal class groups; and she used this in [16] for approximating the wild kernel more closely. But, unexpectedly, the 2-rank of this restricted logarithmic class group C̃ℓ_{F}^{res} sometimes differs from the 2-rank of the group WK_{2}(F). Moreover, in this case the wild kernel WK_{2}(F) may differ from its subgroup K_{2}^{∞}(F) := ∩_{n≥1}K_{2}^{n}(F) of infinite height elements in K_{2}(F). This was observed by J. Tate and then made more explicit by K. Hutchinson (cf [6, 7]).

∗J. Symbolic Comp. (2008)

That last difficulty was finally solved in [12],where the authors constructed a positive class group ad hoc Cℓ_{F}^{pos} which has the same 2-rank as the wild kernel WK_{2}(F). Nevertheless, in case the set PE_{F} of dyadic exceptional primes of the number field F is empty, that group Cℓ_{F}^{pos} appears as a factor of the full narrow logarithmic class group Cℓ_{F}^{res} (without any assumption on the degree), so one may still use narrow logarithmic classes in order to compute the 2-rank of the wild kernel.

In the present paper we use the results from [3] on (ordinary) logarithmic class groups and develop an algorithm for computing the narrow groups C̃ℓ_{F}^{res} in arbitrary number fields F. As a consequence, this algorithm calculates the 2rank of the wild kernel WK_{2}(F) whenever the field F has no dyadic exceptional places.

The computation of the 2-rank of WK_{2}(F) in the remaining case (PE_{F} ≠ ∅) will be solved in a forthcoming article where we compute the finite positive classgroup Cℓ_{F}^{pos} and its subgroup C̃ℓ_{F}^{pos} of positive classes of degree 0.

2 The group of narrow logarithmic classes C̃ℓ_{F}^{res}

In this preliminary section we recall the definition and the main properties of the arithmetic of restricted (or narrow) logarithmic classes. We refer to [9] and [16] for a more detailed account.

Throughout this paper the prime number ℓ equals 2 and F is a number field of degree n = r + 2c with r real places, c complex places and d dyadic places.

According to [8], for every finite place 𝔭 of F there exists a 2-adic 𝔭-valuation ṽ_{𝔭} which is related to the wild 𝔭-symbol in case the cyclotomic ℤ_{2}-extension of F_{𝔭} contains i. The degree deg_{F} 𝔭 of the place 𝔭 is a 2-adic integer such that the image of R_{F} := ℤ_{2} ⊗_{ℤ} F^{×} under the map Log | |_{𝔭} is the ℤ_{2}-module deg_{F} 𝔭 ℤ_{2} (see [9]), where Log denotes the usual 2-adic logarithm and | |_{𝔭} is the 2-adic absolute value at the place 𝔭.

The construction of the 2-adic logarithmic valuations ṽ_{𝔭} yields:

∀α ∈ R_{F} := ℤ_{2} ⊗_{ℤ} F^{×} : ∑_{𝔭∈P l_{F}^{0}} ṽ_{𝔭}(α) deg_{F} 𝔭 = 0 ,  (1)

where P l_{F}^{0} is the set of finite places of the number field F. Setting

d̃iv_{F} (α) := ∑_{𝔭∈P l_{F}^{0}} ṽ_{𝔭}(α) 𝔭

with values in Dℓ_{F} := ⨁_{𝔭∈P l_{F}^{0}} ℤ_{2} 𝔭, we obtain by ℤ_{2}-linearity:

deg_{F} (d̃iv_{F} (α)) = 0 .  (2)

We then define the subgroup of logarithmic divisors of degree 0 by:

D̃ℓ_{F} := {𝔞 = ∑_{𝔭∈P l_{F}^{0}} a_{𝔭}𝔭 ∈ Dℓ_{F} | deg_{F} 𝔞 := ∑_{𝔭∈P l_{F}^{0}} a_{𝔭} deg_{F} 𝔭 = 0};

and the group of principal logarithmic divisors as the image of R_{F} by d̃iv_{F}:

P̃ℓ_{F} := {d̃iv_{F} (α) | α ∈ R_{F} } .

Because of (2), P̃ℓ_{F} is a subgroup of D̃ℓ_{F}. And by the so-called extended Gross conjecture the factor group

C̃ℓ_{F} := D̃ℓ_{F} /P̃ℓ_{F}

is a finite group, the 2-group of logarithmic divisor classes (of degree 0) of the field F introduced in [9].

Now let PR_{F} := {𝔭_{1}, …, 𝔭_{r}} be the (non empty) set of real places of the field F and F^{+} be the subgroup of all totally positive elements in F^{×}, i.e. the kernel of the sign map

sign_{F}^{∞} : F^{×} → {±1}^{r}

which maps x ∈ F onto the vector of the signs of the real conjugates of x. For

P̃ℓ_{F}^{+} := {d̃iv_{F} (α) | α ∈ R_{F}^{+} := ℤ_{2} ⊗_{ℤ} F^{+}}

the factor group

C̃ℓ_{F}^{res} := D̃ℓ_{F} /P̃ℓ_{F}^{+}

is the 2-group of narrow (or restricted) logarithmic divisor classes (of degree 0) introduced in [15]; and it is also finite under the extended Gross conjecture.

In order to make it more suitable for actual computations, we may define it in a slighty different way by introducing real signed divisors of degree 0:

Definition 1. With the notations above, the 2-group of real signed logarithmic divisors (of degree 0) is the direct sum:

D̃ℓ_{F}^{res} := D̃ℓ_{F} ⊕ {±1}^{r};

and the subgroup of principal real signed logarithmic divisors is the image:

P̃ℓ_{F}^{res} := {(d̃iv_{F} (α), sign_{F}^{∞}(α)) | α ∈ R_{F} }

of R_{F} := ℤ_{2} ⊗_{ℤ} F^{×} under the (d̃iv_{F} , sign_{F}^{∞}) map. The factor group:

C̃ℓ_{F}^{res} := D̃ℓ_{F}^{res}/P̃ℓ_{F}^{res}

is the 2-group of narrow logarithmic divisor classes (of degree 0).

Because of the weak approximation theorem, every class in C̃ℓ_{F}^{res} can be represented by a pair (𝔞, 1) where the vector 1 has all entries 1. So the canonical map 𝔞 ↦ (𝔞, 1) induces a morphism from D̃ℓ_{F} onto C̃ℓ_{F}^{res}, the kernel of which is P̃ℓ_{F}^{+}. We conclude as expected:

C̃ℓ_{F}^{res} = D̃ℓ_{F}^{res}/P̃ℓ_{F}^{res} ≃ D̃ℓ_{F} /P̃ℓ_{F}^{+}.

We are now in a situation to present an algorithm for computing narrow logarithmic classes. It uses our previous results of [3] on (ordinary) logarithmic classes and mimics the classical feature concerning narrow and ordinary ideal classes. We note that this algorithm is a bit more intricate in the logarithmic context since the logarithmic units are not algebraic numbers and are therefore not exactely known from a numerical point of view.

3 The algorithm for computing C̃ℓ_{F}^{res}

We assume in this section that the number field F has at least one real place and that the logarithmic 2-class group C̃ℓ_{F} is isomorphic to the sum

C̃ℓ_{F} ≃ ⨁_{j=1}^{ν} ℤ/2^{n_{j}}ℤ

subject to 1 ≤ n_{1} ≤ ... ≤ n_{ν}. Let 𝔞_{j} (1 ≤ j ≤ ν) be fixed representatives of the ν generating divisor classes (of degree 0). We let (ϵ_{i})_{i=1,...,r} denote the canonical basis of the multiplicative 𝔽_{2}-space {±1}^{r}.

Thus any real signed divisor (𝔞, ϵ) in D̃ℓ_{F}^{res} can be uniquely written:

(𝔞, ϵ) = (∑_{j=1}^{ν} a_{j}𝔞_{j} + d̃iv_{F} (α), ∏_{i=1}^{r} ϵ_{i}^{b_{i}} sign_{F}^{∞}(α))

with suitable integers a_{j} ∈ ℤ, b_{i} ∈ {0, 1} and α ∈ R_{F}.

Then the (𝔞_{j}, 1)_{j=1,...,ν} together with the (0, ϵ_{i})_{i=1,...,r} are a finite set of generators of the narrow class group C̃ℓ_{F}^{res}. And we just need to detect the relations among those.

From the description of the logarithmic class group C̃ℓ_{F} above we get:

2^{n_{j}}𝔞_{j} = d̃iv_{F} (α_{j}),

with α_{j} ∈ R_{F} for j = 1, …, ν. So we can define coefficients c_{ν+i,j} in {0, 1} by:

sign_{F}^{∞}(α_{j}) = ((−1)^{c_{ν+1,j}}, …, (−1)^{c_{ν+r,j}})

Consequently, a first set of relations is given by the columns of the following matrix A ∈ ℤ_{2}^{(ν+r)×ν}:

A = ( \begin{matrix}2^{n_{1}} & 0 & ⋯ & 0 & 0 \\ 0 & 2^{n_{2}} & ⋯ & 0 & 0 \\ .. & . & ⋯ & . & . \\ .. & . & ⋯ & . & . \\ 0 & 0 & ⋯ & 2^{n_{ν−1}} & 0 \\ 0 & 0 & ⋯ & 0 & 2^{n_{ν}} \\ −− & −− & − − − & −− & −− \\ & & c_{i,j} \end{matrix} ) .

Now, the ν elements α_{j} are only given up to logarithmic units. Hence, we must additionally consider the sign-function on the 2-group Ẽ_{F} of logarithmic units of F (see [8]). More precisely, in case

Ẽ_{F} = {±1}× < ε̃_{1}, ..., ε̃_{r+c} > ,  (3)

we define exponents b_{i,j} via

sign_{F}^{∞}(ε̃_{j}) = ∏_{i=1}^{r} ϵ_{i}^{b_{i,j}}  (4)

and we have, of course:

sign_{F}^{∞}(−1) = ∏_{i=1}^{r} ϵ_{i} .

From a computational point of view things are a bit more complicated. We just know that

Ẽ_{F} = {x ∈ R_{F} | ∀𝔭 : ṽ_{𝔭}(x) = 0}  (5)

is a subgroup of the 2-group of 2-units E′_{F}. If we assume that there are exactly d places 𝔭_{1}, ..., 𝔭_{d} containing 2 in F, we have, say,

E′_{F} = {±1} × ⟨ε_{1}, ..., ε_{r+c+d−1}⟩ .

In the same way that in [3], for the calculation of Ẽ_{F} we fix a precision e for our 2-adic approximations by requiring for elements ε of E′_{F} the relation

ṽ_{𝔭_{i}}(ε) ≡ 0 mod 2^{e} (1 ≤ i ≤ d) .

We obtain a system of generators of Ẽ_{F} by computing the nullspace of the matrix

M = ( \begin{matrix}& | & 2^{e} & ⋯ & 0 \\ ṽ_{𝔭_{i}}(ε_{j}) & | & · & ⋯ & · \\ & | & 0 & ⋯ & 2^{e} \end{matrix} )

with r + c + 2d − 1 columns and d rows. We assume that the nullspace is generated by the columns of the matrix

M′ = ( \begin{matrix}& C \\ − & − & − \\ & D \end{matrix} )

where C has r + c + d − 1 and D exactly d rows. It suffices to consider C. Each column (n_{1}, ..., n_{r+c+d−1})^{tr} of the matrix C corresponds to a unit

∏_{i=1}^{r+c+d−1} ε_{i}^{n_{i}} ∈ Ẽ_{F} R_{F}^{2^{e}}

so that we can choose

ε̃:= ∏_{i=1}^{r+c+d−1} ε_{i}^{n_{i}}

as an approximation for a logarithmic unit. This procedure yields k ≥ r + c logarithmic units. Of course, by the generalized Gross conjecture we would have exactly r + c such units.

If the integer k which we get in our calculations is not much larger than r +c then we will proceed with the k generating elements of Ẽ_{F} obtained. Otherwise, we reduce the number of generators by computing a basis of the submodule of ℤ^{r+c+d−1} which is the span of the columns of C. Hence, from now on we may assume that we have determined exactly r + c generators ε̃_{1}, ..., ε̃_{r+c} of Ẽ_{F}.

To conclude, with the notations of (4) the columns of the following matrix R ∈ ℤ_{2}^{(ν+r)×(ν+2r+c+1)} generate all relations for the (𝔞_{j}, 1) and the (0, ϵ_{j}):

R = ( \begin{matrix}2^{n_{1}} & & & | & 0 & ⋯ & 0 & | & 0 & ⋯ & 0 & 0 \\ & ... & & | & ⋮ & & ⋮ & | & ⋮ & & ⋮ & ⋮ \\ & & 2^{n_{ν}} & | & 0 & ⋯ & 0 & | & 0 & ⋯ & 0 & 0 \\ −− & −− & − − − & | & −− & −− & −− & | & −− & −− & −− & −− \\ & & & | & 2 & & & | & b_{1,1} & ⋯ & b_{1,r+c} & 1 \\ & c_{i,j} & & | & & ... & & | & ⋮ & & ⋮ & ⋮ \\ & & & | & & & 2 & | & b_{ν,1} & ⋯ & b_{ν,r+c} & 1 \end{matrix} )

4 Applications in K-Theory

We adopt the notations and definitions in this section from [12]. In particular i denotes a primitive fourth root of unity; and we say that the number field F is exceptional when i is not contained in the cyclotomic ℤ_{2}-extension F^{c} of F, i.e. whenever the cyclotomic extension F^{c}[i]/F is not procyclic.

We say that a non-complex place 𝔭 of a number field F is signed whenever the local field F_{𝔭} does not contain the fourth root of unity i. These are the places which do not decompose in the extension F [i]/F. For such a place 𝔭, there exists a non trivial sign-map

sign_{𝔭} : F_{𝔭}^{×} → {±1},

given by the Artin reciprocity map F_{𝔭}^{×} → Gal(F_{𝔭}[i]/F_{𝔭}) of class field theory.

We say that a non-complex place 𝔭 of F is logarithmically signed if and only if one has i ∉ F_{𝔭}^{c}. These are the places which do not decompose in F^{c}[i]/F^{c}. So the finite set PLS_{F} of logarithmic signed places of the field F only contains:

(i) the subset PR_{F} of infinite real places and

(ii) the subset PE_{F} of exceptional dyadic places, i.e. the set of logarithmic signed places above the prime 2.

We say that a non-complex place 𝔭 of F is logarithmically primitive if and only if 𝔭 does not decompose in the first step E/F of the cyclotomic ℤ_{2}-extension F^{c}/F. Finally we say that an exceptional number field F is primitive whenever there exists an exceptional dyadic place which is logarithmically primitive.

Naturally, the task arises to determine logarithmically signed places, i.e. those non complex places of F for which i is not contained in F_{𝔭}^{c}:

Proposition 2. Let E_{𝔭} be the first quadratic extension of F_{𝔭} in the tower of field extension from F_{𝔭} to F_{𝔭}^{c}. Then i ∈ F_{𝔭}^{c} holds precisely for i ∈ E_{𝔭}.

Proof. Since F_{𝔭}^{c}/F_{𝔭} is a ℤ_{2}-extension, it contains exactly one quadratic extension E_{𝔭} of F_{𝔭}. So we immediately obtain:

i ∈ F_{𝔭}^{c} ⇔ F_{𝔭}(i) ⊆ E_{𝔭} ⇔ i ∈ E_{𝔭} .

Remark. The extension E_{𝔭} is F_{𝔭}(α_{k}) where k is the smallest integer such that α_{k} does not belong to F_{𝔭} with α_{0} = 0 and α_{k+1} = \sqrt{2 + α_{k}}.

We assume in the following that the number field F has no exceptional dyadic place. Let us introduce the group Cℓ_{F}^{res} of narrow logarithmic classes without any assumption of degree:

Cℓ_{F}^{res} = Dℓ_{F}^{res}/P̃ℓ_{F}^{res}.

Via the degree map, we obtain the direct decomposition:

Cℓ_{F}^{res} ≃ ℤ_{2} ⊕ C̃ℓ_{F}^{res},

where the torsion subgroup C̃ℓ_{F}^{res} was yet computed in the previous section. So the quotient of exponent 2

2Cℓ_{F}^{res} := Cℓ_{F}^{res}/(Cℓ_{F}^{res})^{2}

contains both as hyperplanes the two quotients ^{2}C̃ℓ_{F}^{res} relative to C̃ℓ_{F}^{res} and 2Cℓ_{F}^{pos} relative to the positive class group Cℓ_{F}^{pos} introduced in [12].

Since, according to [12], this later gives the 2-rank of the wild kernel WK_{2}(F). we can extend the results of K. Hutchinson [6, 7] as follows:

Theorem 3. Let F be a number field which has no exceptional dyadic places.

(i) If F is not exceptional (i.e. in case i ∈ F^{c}) the wild kernel WK_{2}(F) coincide with the subgroup K_{2}^{∞}(F) = ∩_{n≥1}K_{2}^{n}(F) of infinite height elements in K_{2}(F); the group C̃ℓ_{F}^{res} of narrow logarithmic classes coincide with the group C̃ℓ_{F} of (ordinary) logarithmic classes; and one has immediately:

rk_{2} WK_{2}(F) = rk_{2} K_{2}^{∞}(F) = rk_{2} C̃ℓ_{F}^{res} = rk_{2} C̃ℓ_{F}

(ii) If F is exceptional ( i.e. in case i ∉ F^{c}) the subgroup K_{2}^{∞}(F) has index 2 in the wild kernel WK_{2}(F) and one still has:

rk_{2} WK_{2}(F) = rk_{2} C̃ℓ_{F}^{res} ≥ 1

(ii,a) In case WK_{2}(F) and K_{2}^{∞}(F) have the same 2-rank, this gives:

rk_{2} K_{2}^{∞}(F) = rk_{2} C̃ℓ_{F}^{res} ≥ 1.

(ii,b) And in case K_{2}^{∞}(F) is a direct summand in WK_{2}(F), one has:

rk_{2} K_{2}^{∞}(F) = rk_{2} C̃ℓ_{F}^{res} − 1.

Proof. In the non exceptional case, the number field F is not locally exceptional, i.e. has no logarithmic signed places: PE_{F} = PR_{F} = ∅. In particular, narrow logarithmic classes coincide with ordinariry logarithmic classes and the result follows from [12].

In the exceptional case, the number field F may have real places, so the narrow logarithmic class group C̃ℓ_{F}^{res} may differ from the ordinary logarithmic class group. Moreover, because the assumption P E_{F} = ∅ and accordingly to Hutchinson [6, 7], the subgroup K_{2}^{∞}(F) has index 2 in the wild kernel WK_{2}(F).

Remark. It remains to determine whether a number field F is not exceptional, i.e. whether the cyclotomic ℤ_{2} extension F^{c} contains the fourth root of unity i. Of course if i ∈ F^{c} then i is contained in the quadratic subfield E/F in F^{c}. Now the finite subfields of ℚ^{c} are the real cyclic fields ℚ^{(s)} = ℚ[ζ_{2^{s+2}} +ζ_{2^{s+2}}^{−1}] and the finite extensions of F^{c} are of the form F ℚ^{(s)}. So we only need to check whether i is contained in F [ζ_{2^{s+2}} + ζ_{2^{s+2}}^{−1}] where s is minimal with ζ_{2^{s+2}} + ζ_{2^{s+2}}^{−1} ∉ F.

5 Examples

The methods described here are implemented in the computer algebra system Magma [2]. Many of the fields used in the examples were results of queries to the QaoS number field database [13, section 6].

The wild kernel W K_{2}(F) is contained in the tame kernel K_{2}(O_{F}). Let μ(F) be the order of the torsion subgroup of F^{×} and for a prime 𝔭 of F over p denote by μ^{1}(F_{𝔭}) the p-Sylow subgroup of the torsion subgroup of F_{𝔭}^{×}. By coupling Moore’s exact sequence and the localization sequence [5, section 1] one obtains the index formula [1, equation (6)]:

(K_{2}(O_{F}) : W K_{2}(F)) = \frac{2^{r}}{|μ(F)|} ∏_{𝔭} |μ^{1}(F_{𝔭})|,

where 𝔭 runs through all finite places and r is the number of real places of F.

We apply this in the determination of the structure of WK_{2}(F) in the cases where the structure of K_{2}(O_{F}) is known.

Abelian groups are given as a list of the orders of their cyclic factors;

[:] denotes the index (K_{2}(O_{F}) : WK_{2}(F));

d_{F} denotes the discriminant for a number field F;

Cℓ_{F} denotes the class group, P the set of dyadic places;

Cℓ′_{F} denotes the 2-part of Cl/⟨P ⟩;

C̃ℓ_{F} denotes the logarithmic classgroup;

Cℓ_{F}^{res} denotes the group of narrow logarithmic classes;

rk_{2} denotes the 2-rank of the wild kernel WK_{2};

WK_{2} denotes the wild kernel in K_{2}(F);

K_{2}^{∞} denotes the subgroup of infinite height elements in K_{2}(F).

5.1 Imaginary quadratic fields

K. Belabas and H. Gangl have developed an algorithm for the computation of the tame kernel K_{2}O_{F} [1]. The following table contains the structure of K_{2}O_{F} as computed by Belabas and Gangl and the 2-rank of the wild kernel WK_{2}(F) calculated with our methods. We also give the structure of the wild kernel if it can be deduced from the structure of K_{2}O_{F} and of the rank of the wild kernel computed here or in [14]. The structure of the tame kernel K_{2}(O_{F}) of all fields except for the starred entries has been proven by Belabas and Gangl.

The table gives the structure of the wild kernel of all imaginary quadratic fields F with no exceptional places and discriminant |d_{F} | < 1000.

\begin{matrix}d_{F} & Cℓ_{F} & K_{2}(O_{F}) & [:] & Cℓ′_{F} & C̃ℓ_{F} & C̃ℓ_{F}^{res} & rk_{2} & WK_{2} & K_{2}^{∞} \\ -68 & [ 4 ] & [ 8 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [4] & [2] \\ -132 & [2,2] & [ 4 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -136 & [ 4 ] & [ 4 ] & 1 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [4] & [2] \\ -164 & [ 8 ] & [ 4 ] & 2 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -228 & [2,2] & [ 12] & 6 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -260 & [2,4] & [ 4 ] & 2 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -264 & [2,4] & [ 6 ] & 3 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -292 & [ 4 ] & [ 4 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -328 & [ 4 ] & [ 2 ] & 1 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -356 & [ 12 ] & [ 4 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -388 & [ 4 ] & [ 8 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [4] & [2] \\ -420 & [2,2,2] & [2,4] & 2 & [2,2] & [2,2] & [2,2] & 2 & [2,2] & [2] \\ -452 & [ 8 ] & [ 8 ] & 2 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [4] & [2] \\ -456 & [2,4] & [ 2 ] & 1 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -516 & [2,6] & [ 12 ] & 6 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -520 & [2,2] & [ 2 ] & 1 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -548 & [ 8 ] & [ 4 ] & 2 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -580 & [2,4] & [ 4 ] & 2 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -584 & [ 16 ] & [ 2 ] & 1 & [ 8 ] & [ 8 ] & [ 8 ] & 1 & [2] & [ ] \\ -644 & [2,8] & [2,16] & 2 & [2,4] & [2,4] & [2,4] & 2 & [2,8] & [?] \\ -708 & [2,2] & [ 4 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -712 & [ 8 ] & [ 2 ] & 1 & [ 4 ] & [ 4 ] & [ 4 ] & 1 & [2] & [ ] \\ -740 & [2,8] & [ 4 ] & 2 & [ 8 ] & [ 8 ] & [ 8 ] & 1 & [2] & [ ] \\ -772 & [ 4 ] & [ 8 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [4] & [2] \\ -776 & [ 20 ] & [ 4 ] & 1 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [4] & [2] \\ -804 & [2,6] & [ 36 ] & 6 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [6] & [3] \\ -836 & [2,10] & [ 4 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \\ -840 & [2,2,2] & [2,6] & 3 & [2,2] & [2,2] & [2,2] & 2 & [2,2] & [2] \\ -868 & [2,4] & [2,4] & 2 & [2,2] & [2,2] & [2,2] & 2 & [2,2] & [2] \\ -904 & [ 8 ] & [4] & 1 & [ 4 ] & [ 4 ] & [ 4 ] & 2 & [4] & [2] \\ -964 & [ 12 ] & [ 8 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [4] & [2] \\ -996 & [2,6] & [ 4 ] & 2 & [ 2 ] & [ 2 ] & [ 2 ] & 1 & [2] & [ ] \end{matrix}

5.2 Real Quadratic Fields

The following table contains all real quadratic fields F with no exceptional places and discriminant |d_{F} | < 1000. All these quadratic fields are exceptional.

\begin{matrix}d_{F} & Cℓ_{F} & [:] & |P | & |PE| & Cℓ′_{F} & C̃ℓ_{F} & C̃ℓ_{F}^{res} & rk_{2} \\ 28 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 56 & [ ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 60 & [ 2 ] & 24 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 92 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 120 & [ 2 ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 124 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 156 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 184 & [ ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 188 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 220 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 248 & [ ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 284 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 312 & [ 2 ] & 12 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 316 & [ 3 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 348 & [ 2 ] & 24 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 376 & [ ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 380 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 412 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 440 & [ 2 ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 444 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 476 & [ 2 ] & 8 & 1 & 0 & [ 2 ] & [ 2 ] & [ 2, 2 ] & 2 \\ 604 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 632 & [ ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 636 & [ 2 ] & 24 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 668 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 696 & [ 2 ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 732 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 760 & [ 2 ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 764 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 796 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 824 & [ ] & 4 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 860 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 888 & [ 2 ] & 12 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 892 & [ 3 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 924 & [ 2, 2 ] & 24 & 1 & 0 & [ 2 ] & [ 2 ] & [ 2, 2 ] & 2 \\ 952 & [ 2 ] & 4 & 1 & 0 & [ 2 ] & [ 2 ] & [ 2, 2 ] & 2 \\ 956 & [ ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \\ 988 & [ 2 ] & 8 & 1 & 0 & [ ] & [ ] & [ 2 ] & 1 \end{matrix}

The following table contains extensions with class number 32 up to discriminant 222780 and extensions with class number 64 up tp discriminant 805596.

\begin{matrix}d_{F} & Cℓ_{F} & [:] & |P | & |PE| & Cℓ′_{F} & C̃ℓ_{F} & C̃ℓ_{F}^{res} & rk_{2} \\ 112924 & [ 2,16 ] & 8 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 120796 & [ 2,16 ] & 8 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 136120 & [ 2,16 ] & 4 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 153660 & [2,2,8] & 8 & 1 & 0 & [ 2,8 ] & [ 2,8 ] & [2,2,8] & 3 \\ 158844 & [2,2,8] & 8 & 1 & 0 & [ 2,8 ] & [ 2,8 ] & [2,2,8] & 3 \\ 163576 & [ 2,16 ] & 4 & 1 & 0 & [ 2,8 ] & [ 2,8 ] & [2,2,8] & 3 \\ 170872 & [ 2,16 ] & 4 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 176316 & [ 2,16 ] & 24 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 176440 & [ 2,16 ] & 4 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 196540 & [ 2,16 ] & 8 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 202524 & [ 2,16 ] & 24 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 207480 & [2,2,2,4] & 4 & 1 & 0 & [2,2,4] & [2,2,4] & [2,2,2,4] & 4 \\ 213180 & [2,2,2,4] & 24 & 1 & 0 & [2,2,4] & [2,2,4] & [2,2,2,4] & 4 \\ 221276 & [ 2,16 ] & 8 & 1 & 0 & [ 16 ] & [ 16 ] & [ 2,16 ] & 2 \\ 222780 & [2,2,8] & 8 & 1 & 0 & [ 2,8 ] & [ 2,8 ] & [2,2,8] & 3 \\ 374136 & [2,2,16] & 12 & 1 & 0 & [ 2,16 ] & [ 2,16 ] & [2,2,16] & 3 \\ 382204 & [ 2,32 ] & 8 & 1 & 0 & [ 32 ] & [ 32 ] & [ 2,32 ] & 2 \\ 449436 & [2,2,16] & 8 & 1 & 0 & [ 2,16 ] & [ 2,16 ] & [2,2,16] & 3 \\ 484764 & [2,2,16] & 24 & 1 & 0 & [ 2,16 ] & [ 2,16 ] & [2,2,16] & 3 \\ 506940 & [2,2,2,8] & 24 & 1 & 0 & [2,2,8] & [2,2,8] & [2,2,2,8] & 4 \\ 805596 & [2,2,16] & 24 & 1 & 0 & [ 2,16 ] & [ 2,16 ] & [2,2,16] & 3 \end{matrix}

5.3 Biquadratic Extensions

The following table contains quadratic and biquadratic number fields. The biquadratic fields are the compositum of the the first quadratic extensions and one of the other quadratic extensions. All fields are exceptional.

F  d_{F} r  Cℓ_{F}  [:] |P | Cℓ′_{F}  C̃ℓ_{F}  C̃ℓ_{F}^{res}  rk_{2}

K  9660 2 [2,2,2]  8 1 [ 2,2 ] [ 2,2 ] [ 2,2,2 ]  3
L_{1}  9340 2 [ 10 ]  8 1  [ ]  [ ]  [ 2 ]  1
KL_{1}  4 [2,2,2,10] 128 2 [2,2,2] [2,2,2] [2,2,2,2,2] 5
L_{2} 13020 2 [2,2,2] 24 1 [ 2,2 ] [ 2,2 ] [ 2,2,2 ]  3
KL_{2}  4 [2,2,4] 384 2 [ 2,2 ] [2,2,2] [2,2,2,2,2,2] 6
L_{3} 15708 2 [2,2,2] 8 1 [ 2,2 ] [ 2,2 ] [ 2,2,2 ] 3
KL_{3}  4 [2,2,4,28] 128 2 [ 2,2,4] [2,2,4] [2,2,2,2,2,4] 6

5.4 Examples of higher degrees

f  d_{F} r Gal  Cℓ_{F}  [:] |P | Cℓ′_{F}  C̃ℓ_{F} C̃ℓ_{F}^{res} rk_{2}

x^{4} + x^{2} − 6x + 1  -3312 2 D(4)  [ ]  8  1 [ ]  [ ] [ 2 ] 1
x^{4} − 2x^{3} − 2x^{2} + 18x + 21  3600 0 E(4)  [ 2 ]  6  1 [ 2 ] [ 2 ] [ 2 ] 1
x^{4} + 25x^{2} + 400  608400 0 E(4) [ 4,8 ] 72 2 [ 4 ] [ 4 ] [ 4 ] 1
x^{4} + 56x^{2} − 32x + 713 700672 0 D(4) [2,2,4] 4 1 [2,4] [2,4] [2,4] 2
x^{4} + 3x^{2} − 30x + 66  723600 0 D(4) [ 4,8 ] 24 2 [ 2 ] [ 2 ] [ 2 ] 1
x^{4} − 2x^{3} + 29x^{2} − 28x + 417  781456 0 E(4) [ 4,8 ]  2  1 [2,8] [2,8] [2,8] 2
x^{4} + 10x^{2} − 28x + 18  815360 0 D(4) [ 4,8 ]  4  1 [2,4] [ 4 ] [ 4 ] 1
x^{4} + 12x^{2} − 40x + 81  825600 0 D(4) [ 4,8 ]  1  1 [ 8 ] [ 8 ] [ 8 ] 1

x^{6}+ 2x^{5}−4x^{4}−16x^{3}+6x^{2}+44x+308 -6832605533873152 0 S(6) [ 2,2 ]  8  2 [ 2 ] [ 2 ] [ 2 ] 1
x^{6} − 2x^{4} + 10x^{2} + 12x + 260  -3797563908766976 0 S(6) [ 2,2 ]  8  2 [ 2 ] [ 2 ] [ 2 ] 1
x^{6} + 2x^{5} + 4x^{4} − 2x^{2} − 4x + 260  -382132112360448 0 S(6)  [ 4 ]  48 2 [ 4 ] [ 4 ] [ 4 ] 1
x^{6} − 26x^{4} − 16x^{3} + 90x^{2} − 52x + 68 -212547578875136 4 S(6)  [ 2 ] 128 2 [ ]  [ ] [2,2] 2
x^{6} − 7x^{4} + 14x^{2} − 7  1075648 6 C(6)  [ ]  128 1 [ ]  [ ] [ 2 ] 1

x^{8} + 4x^{7} −8x^{6} − 42x^{5} + 11x^{4}

+130x^{3} +15x^{2}−106x+11 8090338299904 8 [A(4)^{2}]2 [ ] 512 2 [ ] [ ] [2,2] 2

x^{8}−2x^{7}−27x^{6}+62x^{5}+185x^{4}

−520x^{3}−40x^{2}+832x−496 9082363580416 8 E(8) [ ] 512 2 [ ] [ ] [ 2 ] 2

x^{8}−4x^{7}−20x^{6}+30x^{5}+105x^{4}

−30x^{3}−168x^{2}−78x−3 9299377062144 8 [A(4)^{2}]2 [ ] 2048 2 [ ] [ ] [ 2 ] 1

x^{8}+2x^{7}−22x^{6}−8x^{5}+159x^{4}

−160x^{3}−110x^{2}+186x−47  9451049953536 8 [A(4)^{2}]2 [ ]  2048 2 [ ]  [ ] [2,2] 2

References

[1] K. Belabas & H. Gangl, Generators and Relations for K_{2}O_{F}, K-Theory 31 2004, 195–231.

[2] J.J. Cannon et al., The computer algebra system Magma, The University of Sydney

(2006), http://magma.maths.usyd.edu.au/magma/.

[3] F. Diaz y Diaz, J.-F. Jaulent, S. Pauli, M.E. Pohst & F. Soriano, A new algorithm for the computation of logarithmic class groups of number fields, Experimental. Math. 14 (2005), 67–76.

[4] F. Diaz y Diaz & F. Soriano, Approche algorithmique du groupe des classes logarithmiques, J. Number Theory 76 (1999), 1–15.

[5] G. Gras, Remarks on K_{2} of number fields, J. Number Theory 23(3) (1986), 322–335.

[6] K. Hutchinson, The 2-Sylow subgroup of the wild kernel of exceptional number fields, J. Number Th. 87 (2001), 222–238.

[7] K. Hutchinson, On tame and wild kernels of special number fields, J. Numb. Th. 107 (2004), 222–238.

[8] J.-F. Jaulent, Sur le noyau sauvage des corps de nombres, Acta Arithmetica 67 (1994), 335–348.

[9] J.-F. Jaulent, Classes logarithmiques des corps de nombres, J. Théor. Nombres Bordeaux 6 (1994), 301–325.

[10] J.-F. Jaulent & A. Michel, Approche logarithmique des noyaux étales sauvages des corps de nombres, J. Numb. Th. 120 (2006), 72–91.

[11] J.-F. Jaulent & F. Soriano-Gafiuk, Sur le noyau sauvage des corps de nombres et le groupe des classes logarithmiques, Math. Z. 238 (2001), 335–354.

[12] J.-F. Jaulent & F. Soriano-Gafiuk, 2-groupe des classes positives d’un corps de nombres et noyau sauvage de la K-théorie, J. Number Th. 108 (2004), 187–208.

[13] S. Freundt, A. Karve, A. Krahmann, S. Pauli, KASH: Recent Developments, in Mathematical Software - ICMS 2006, Second International Congress on Mathematical Software, LNCS 4151, Springer, Berlin, 2006, http://www.math.tu-berlin.de/˜ kant.

[14] S. Pauli & F. Soriano-Gafiuk, The discrete logarithm in logarithmic ℓ-class groups ands its applications in K-Theory, in Algorithmic number theory, 367–378, Lecture Notes in Comput. Sci. 3076, Springer, 2004.

[15] F. Soriano, Classes logarithmiques au sens restreint, Manuscripta Math. 93 (1997), 409–420.

[16] F. Soriano-Gafiuk, Sur le noyau hilbertien d’un corps de nombres, C. R. Acad. Sci. Paris 330, Série I (2000), 863–866.

Jean-François Jaulent Université Bordeaux I Institut de Mathématiques 351, Cours de la Libération 33405 Talence Cedex, France

jaulent@math.u-bordeaux1.fr

Sebastian Pauli

University of North Carolina Department of Mathematics and Statistics Greensboro, NC 27402, USA

                                     s_pauli@uncg.edu

Michael E. Pohst Technische Universität Berlin Institut für Mathematik MA 8-1 Straße des 17. Juni 136 10623 Berlin, Germany

pohst@math.tu-berlin.de

Florence Soriano-Gafiuk

Université de Metz

Département de Mathématiques Ile du Saulcy 57000 Metz, France

                                     soriano@poncelet.univ-metz.fr