Principalization of ideals in Abelian extensions of number fields

Sébastien Bosca

Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33405 TALENCE Cedex, France

With an Appendix by Georges Gras^{1} and Jean-François Jaulent ^{2}

Abstract

We give a self-contained proof of a general conjecture of G. Gras on principalization of ideals in Abelian extensions of a given field L, which was solved by M. Kurihara in the case of totally real extensions L of the rational field ℚ.

More precisely, for any given extension L/K of number fields, in which at least one infinite place of K totally splits, and for any ideal class c_{L} of L, we construct a finite Abelian extension F/K, in which all infinite places totally split, such that c_{L} become principal in the compositum M = LF.

Acknowledgment

The original version of this paper was sent to the journal in December 2004 and the referee returned a very positive report with many corrections to be done. In the meantime, Sébastien Bosca left the University of Bordeaux, for other activities, and it was not possible for him to make the necessary revision.

Since Jean-François Jaulent was his thesis advisor and since Georges Gras was in part at the origin of the subject (from a conjecture given in 1997), one of the Editors proposed to them to finish the work by incorporating the remarks of the referee, and especially to complete, in an Appendix, an important point of the method.

The Editor is pleased to thank Jean-François Jaulent and Georges Gras for their helpful contribution.

1 Villa la Gardette, Chemin Château Gagnière, 38520 Le Bourg d’Oisans, France 2 Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33405 TALENCE Cedex, France.

Preprint submitted to IJNT 15 December 2011

A – INTRODUCTION

When M/L is an Abelian extension of number fields, the problem of knowing which ideals of L become principal in M is difficult, even if M/L is cyclic. Class field theory gives partial answers. For instance, the Artin-Furtwängler theorem states that when M = H_{L} is the Hilbert class field of L, all ideals of L become principal in M; but the other cases are more mysterious.

When M/L is cyclic, we still have no general answer but the problem is easier. The kernel of the natural map j : Cl_{L} → Cl_{M} is partly known as explained below:

At first, cohomology of cyclic groups says that this kernel is a part of Ĥ^{1}(G, E_{M}), where G = Gal(M/L) and E_{M} is the group of units of M; Ĥ^{1}(G, E_{M}) itself is not well known but its order can be deduced from the order of Ĥ^{0}(G, E_{M}) using the Herbrand quotient; then, the order of Ĥ^{0}(G, E_{M}) depends partly on the natural map E_{L} → U_{S}, where U_{S} is the subgroup of the unit idèles of L which is the product of the groups of local units at the places ramified in M/L (S is the set of such ramified places); finally, the map E_{L} → U_{S} depends on the Frobenius’ of the primes of S in the extension L[μ_{h}, E_{L}^{1/h}]/L (where h = [M : L] and where μ_{h} is the group of hth roots of unity). Obviously, this makes sense only if the primes of S are prime to the degree of the extension.

However, even in this cyclic case, Ker(j) is not completely given by these Frobenius’, so that knowing such Frobenius’, we cannot get really more than a lower bound of the order of Ker(j).

This article deals with such techniques, which allow us to prove, for instance, this easy fact: if a cyclic extension M/L is ramified at only one finite place 𝔮 prime to [M : L], then, as soon as the ramification index e_{𝔮} is large enough (precisely, when |Cl_{L}| divides e_{𝔮}), the capitulation kernel Kerj contains at least the class of 𝔮 in Cl_{L}. This can be easily established by studying Ĥ^{1}(G, E_{M}) and may be seen as a particular case of our main theorem.

On the other hand, this article proves a conjecture of Georges Gras and generalizes a result of Masato Kurihara ([1] and [2]), so that the theorem given here is not only an abstract lower bound of Kerj, using cohomology of cyclic groups, Dirichlet–Herbrand theorem on units of number fields, class field theory, and Kummer duality. Note also that we make use here of new asymptotic methods (“take n large enough”, where n is related to the degree [F : K] in a suitable manner).

Note finally that in the theorem below the hypothesis “at least one infinite place of K totally splits in L/K” is necessary: in [1], Georges Gras gives examples of extensions L/K with ideals which do not become principal in the compositum LK^{ab}, where K^{ab} is the maximal Abelian extension of K.

B – MAIN THEOREM AND COROLLARIES

Main Theorem. Let L/K be a finite extension of number fields in which at least one infinite place of K totally splits. There exists a finite Abelian extension F/K, in which all infinite places split totally, such that every ideal of L becomes principal in the compositum M = LF.

Corollary 1 (K = ℚ). If L is a totally real number field, any ideal of L principalizes in a real cyclotomic extension of L (i.e., in the compositum of L with a real subfield of a suitable cyclotomic extension of ℚ).

Corollary 1 was proved by Masato Kurihara in [2].

Notation: In what follows, K^{ab} denotes the maximal Abelian extension of K and K_{+}^{ab} its maximal totally real subextension (i.e., the subextension of K^{ab}/K fixed under the decomposition groups of the infinite places of K).

Definition: Let us say that a number field N is principal when Cl_{N} is trivial.

Then:

Corollary 2. (i) Let K be a number field with at least one complex place. Then any field containing K^{ab} is principal. (ii) Let K be a totally real number field. Any field containing K_{+}^{ab}, the Galois closure of which has at least one real place is principal.

Corollary 3. (i) Any totally real field containing ℚ_{+}^{ab} is principal. (ii) Any field containing ℚ(i)^{ab} is principal.

Corollary 3, (i), and (ii) for ℚ(i) or any imaginary quadratic field, was proved by Masato Kurihara ([2], theorem 1.1, p. 35 and theorem A.1, p. 46).

Corollary 4. (i) Let K be a number field with at least one complex place. Then K^{ab} is principal. (ii) Let K be a totally real number field. Any field containing K_{+}^{ab} which is contained in one of the subfields of K^{ab} fixed by a complex conjugation is principal.

Corollary 4 proves a conjecture of Georges Gras (Conjecture (0.5), p. 405 of [1]).

C – PROOFS

I. Proof of the main theorem : preliminaries

To prove the theorem, we fix an ideal 𝔞_{L} of L, and we shall construct a finite Abelian extension F of K, which is totally split at all infinite places, cyclic in most cases, such that 𝔞_{L} principalizes in the compositum LF. Obviously, any ideal 𝔟_{L} of L with the same class in Cl_{L} will become principal in LF as well, so that it is enough to fix the class c_{L} of 𝔞_{L} in Cl_{L}. Now, if (c_{i})_{i} is a finite generating system of Cl_{L}, we will obtain a corresponding set (F_{i})_{i} of extensions, and every ideal of L will principalize in LF, where F is the compositum of the (F_{i})_{i}; this will prove the theorem.

So, in what follows, we fix c_{L} in Cl_{L}, and must find F. As the class group of L is the direct sum of its p-parts, for all prime numbers p, we may assume that the order of c_{L} in Cl_{L} is a power of a prime p. So, c_{L} and p are fixed; Cl_{L} is now the p-part of the class group of K, and H_{L} the maximal p-extension contained in the Hilbert class field of L.

(1) One may assume c_{L} ∈ Cl_{L}^{p^{a}} for an arbitrary given integer a

Definition: Let us call Abelian compositum of the extension L/K any extension N = LF, where F is a finite Abelian extension of K, totally split at all infinite places of K.

One fixes an integer a; in case c_{L} ∉ Cl_{L}^{p^{a}}, one will build an Abelian compositum L′ of L/K, such that the extended class c_{L′} = j(c_{L}) satisfies c_{L′} ∈ Cl_{L′}^{p^{a}}; so, if N′ is an Abelian compositum of L′/K in which c_{L′} principalizes, it is as well an Abelian compositum of L/K, so that one can legitimately replace L by L′, in which case one has c_{L′} ∈ Cl_{L′}^{p^{a}}.

Let’s build such a L′ = LF_{0} as follows: let 𝔮 be a prime of L satisfying the following three conditions:

(i) 𝔮 totally splits in L/ℚ;

(ii) Frob(𝔮, L[μ_{2p^{a}}]/L) = id;

(iii) Frob(𝔮, H_{L}/L) = c_{L}.

If such a 𝔮 exists with say 𝔮|q, the first two conditions imply the existence of a (cyclic) subfield F′_{0} of ℚ(μ_{q}), with degree [F′_{0} : ℚ] = p^{a}, which is totally ramified at the prime q; the first condition implies that the compositums F_{0} = KF′_{0} and L′ = LF_{0} have again a degree p^{a} over K and L, respectively. Now L′/L is totally ramified at 𝔮, say 𝔮 = 𝔮′ ^{p^{a}}, for a prime 𝔮′ in L′; so, according to the third condition, the extended class c_{L′} = j(c_{L}) satisfies:

c_{L′} = \overline{𝔮} = \overline{𝔮′} ^{p^{a}} ∈ Cl_{L′}^{p^{a}}

as expected.

Now we only have to verify the existence of such a prime 𝔮. The three conditions defining 𝔮 all depend on Frob(q, H̃_{L}[μ_{2p^{a}}]/ℚ), where H̃_{L} is the Galois closure of H_{L} over ℚ; the first two conditions are equivalent to the fact that this Frobenius is in the subgroup Gal(H̃_{L}[μ_{2p^{a}}]/L[μ_{2p^{a}}]); so they are compatible with the last condition for any c_{L} in Cl_{L}, if and only if one has: L[μ_{2p^{a}}] ∩ H_{L} = L; if this is right, the Čebotarev theorem states there are infinitely many 𝔮 satisfying the conditions. When this is wrong, we replace L by L′′ = L[μ_{2p^{a}}] ∩ H_{L} which verifies L′′[μ_{2p^{a}}] ∩ H_{L′′} = L′′.

As explained above, this last change is possible in case L′′ = L[μ_{2p^{a}}]∩H_{L} is an Abelian compositum of L/K. In fact, one has L′′ = LF where F is contained in the maximal p-subfield of K[μ_{2p^{a}}], which is clearly Abelian and finite but in which infinite places are maybe not totally split for p = 2.

So, for p = 2, we shall complete the proof, and we take in this particular case L′′ = LK[μ_{2^{b}}]_{+}, where the symbol + denotes the maximal ∞-split subextension over K, and where b is an integer, which is chosen large enough so that L′′ contains H_{L} ∩ LK[μ_{2^{a+1}}]_{+} and L′′/L has degree at least 2.

Suppose we have found a prime 𝔮′′ of L′′ satisfying the following conditions:

(i) 𝔮′′ totally splits in L′′/ℚ;

(ii) Frob(𝔮′′, L′′[μ_{2^{a+1}}]/L′′) = id;

(iii) Frob(𝔮′′, H_{L′′}/L′′) = c_{L′′},

where c_{L′′} is extended from c_{L}. Then, let F′_{0} be the totally real subfield of ℚ[μ_{q}] of degree 2^{a} over ℚ, thus F_{0} = K[μ_{2^{b}}]_{+}F′_{0} and L′′′ = LF_{0} = L′′F′_{0} (which is an Abelian compositum of L/K). If 𝔮′′′ denotes the unique prime of L′′′ above 𝔮′′, the extended class c′′′_{L′′′} in Cl_{L′′′} satisfies the expected condition:

c_{L′′′} = \overline{𝔮′′} = \overline{𝔮′′′} ^{2^{a}} ∈ Cl_{L′′′}^{2^{a}}.

So to conclude we only have to prove the existence of such a prime 𝔮′′ satisfying the three conditions above. But this existence follows from the Čebotarev theorem as soon as the image of c_{L} ∈ Gal(H_{L′′}/L′′) in Gal(H_{L′′} ∩L′′[μ_{2^{a+1}}]/L′′) is trivial.

To check this last point, let us observe that in the class field description the extension of ideal classes j corresponds to the transfer map Ver. Here L′′ contains H_{L} ∩ LK[μ_{2^{a+1}}]_{+}, so H_{L′′} ∩ L′′[μ_{2^{a+1}}] = L′′′ is either L′′ or L′′[i], and the image of j_{L′′/L}(c_{L}) in Gal(L′′′/L′′) is trivial, since one has:

Ver_{B/A→B′/A′}(σ) = σ^{[A′:A]}

when B′/A is Abelian, so:

Res_{L′′′}(Ver_{H_{L}/L→H_{L′′}/L′′}(c_{L})) = Ver_{H_{L}/L→L′′′/L′′}(c_{L}) = c_{L}^{[L′′:L]} = id.

(2) One may assume that L/K is Galois

Let L̃ denotes the Galois closure of L over K. Assume the theorem is proved for L̃/K (in which at least one infinite place totally splits as in L/K). Hence, there exists an Abelian compositum L̃F of L̃/K such that every ideal of L̃ principalize in L̃F; so, c_{L} principalizes in L̃F but maybe does not in LF and we have to study this case.

(a) When c_{L} is norm in L̃/L, say c_{L} = N_{L̃/L}(c̃_{L̃}), the class c̃_{L̃} ∈ Cl_{L̃} principalizes in L̃F, say j_{L̃F/L̃}(c̃_{L̃}) = 1 in Cl_{L̃F}; so we obtain: c_{L} = N_{L̃∩LF/L}(c′_{L̃∩LF}) with c′_{L̃∩LF} = (N_{L̃/(L̃∩LF)}(c̃_{L̃})) and the class c′_{L̃∩LF}, which satisfies j_{LF/L̃∩LF} ∘ N_{L̃/(L̃∩LF)}(c̃_{L̃}) = N_{L̃F/LF} ∘ j_{L̃F/L̃}(c̃_{L̃}) = 1, principalizes in F L; and so does c_{L}.

(b) When c_{L} is not a norm in L̃/L, maybe c_{L} does not principalize in LF. But N_{L̃/L}(ClL̃) contains Cl_{L}^{[L̃:L]} = Cl_{L}^{p^{a}}, where p^{a} is the largest power of p dividing [L̃ : L]. According to Section (1) we replace L by L′ = LF_{0} such that c_{L} ∈ Cl_{L′}^{p^{a}}. Since [L̃′ : L′] = [L̃F_{0} : LF_{0}] divides [L̃ : L], c_{L} is norm in L̃′/L′ and (a) applies.

II. Proof of the theorem : constructing the extension F

(3) The method and a first condition about the prime 𝔮

From now on we suppose L/K is Galois. The prime p and the class c_{L} are fixed and we must build an Abelian compositum LF of L/K such that c_{L} principalizes in LF. For convenience, we choose F/K as a cyclic p-extension, ramified at only one finite place 𝔮 of K, with ramification index e_{𝔮}(F/K) = p^{n} for some integer n. We will see that under some conditions about 𝔮, when n is large enough, c_{L} becomes principal in LF, or in L′F, where L′ is a convenient Abelian compositum of L/K. At the end of the proof, in Section (6), we will study the existence of such a 𝔮 satisfying all conditions.

Now for a given integer n and a given prime 𝔮 of K, we wonder if c_{L} principalizes in M = LF, where F is a cyclic p-extension of K with e_{𝔮}(F/K) = p^{n}, unramified but at 𝔮 and in which all infinite places split totally. The first question is the existence of such an extension F/K. Class field theory gives the answer as follows.

Indeed, N being the maximal Abelian extension of K unramified but at 𝔮 and ∞-split, class field theory describes the Galois group Gal(N/K) from the quotient

J_{K}/K^{×}. ∏_{v|∞} K_{v}^{×}. ∏_{𝔮′≠𝔮}U_{𝔮′} ,

where J_{K} is the idèle group of K and U_{𝔮′} the subgroup of local units of the completion K_{𝔮′} of K at the place 𝔮′. The inertia subgroup of 𝔮 in N/K is, according to class field theory, isomorphic to the quotient

U_{𝔮} / \overline{U_{𝔮} ∩ (K^{×}. ∏_{v|∞} K_{v}^{×}. ∏_{𝔮′≠𝔮}U_{𝔮′})} = U_{𝔮}/\overline{E_{K}} ,

where E_{K} is the group of global units in K and the overlining means topological closure in U_{𝔮} of the diagonal embedding. Of course if F exists, it is contained in the maximal p-extension of N, whose ramification subgroup is the p-part of U_{𝔮}/\overline{E_{K}}, denoted (U_{𝔮}/\overline{E_{K}})_{p}.

We suppose now:

• 𝔮 ∤ p.

So, if F exists, p^{n} divides |(U_{𝔮}/\overline{E_{K}})_{p}|, that is, under the assumption 𝔮 ∤ p:

• μ_{p^{n}} ⊂ K_{𝔮}^{×}, and

• E_{K} ⊂ U_{𝔮}^{p^{n}} .^{3}

These necessary conditions suffice to ensure the existence of F, according to an obvious lemma, which reads: if A is an Abelian finite group and C is a cyclic subgroup of A of order divisible by p^{n}, then there exists a cyclic quotient of A in which the image of C has order p^{n}.

Now we suppose that 𝔮 ∤ p satisfies the two conditions above together with the additional assumption:

• 𝔮 is unramified in L/K.^{4}

So F exists; all primes 𝔮_{L} | 𝔮 are ramified in M/L = LF/L with the same index e_{𝔮} = p^{n}; and we have [M : L] = p^{n+d} for some positive integer d.

(4) Obtaining a large cohomology group Ĥ^{0}(G, E_{M})

3 The canonical embedding of E_{K} in K_{𝔮}^{×} must be contained in U_{𝔮}^{p^{n}} since (U_{𝔮})_{p} is here a cyclic group. 4 In fact, in the sequel Bosca will suppose that 𝔮 is totally split in L/K.

Let G denote the Galois group Gal(M/L), which is cyclic of order p^{n+d}, and G = Gal(L/K).

According to the Dirichlet–Herbrand theorem, the character of the representation ℚ ⊗_{ℤ} E_{L} of G = Gal(L/K) given by the group of global units E_{L} is:

χ_{E_{L}} = ∑_{v|∞} Ind_{D_{v}}^{G}1_{D_{v}} − 1_{G} ,

where 1_{G} is the trivial character of G and 1_{D_{v}} the trivial character of the decomposition subgroup D_{v}.

Since at least one infinite place splits totally in the extension L/K, the character of ℚ ⊗_{ℤ} (E_{L}/μ_{L}E_{K}) satisfies

χ_{E_{L}/μ_{L}E_{K}} ≥ χ_{ℤ[G]} − 1 ,

and we can deduce from this the existence of a map:^{5}

φ : E_{L}/μ_{L}E_{K} ⟶ ℤ

such that, in Hom(E_{L}/μ_{L}.E_{K}, ℤ), φ generates a ℤ[G]-submodule whose character is χ_{ℤ[G]} − 1.

On the other hand, since L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L[μ_{p^{n}}] is a Kummer extension, if 𝔮_{L} is one of the primes of L dividing 𝔮, the Frobenius automorphism

σ = Frob(𝔮_{L}, L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L)

corresponds, in the Kummer duality, to the map:

u ↦ (u^{1/p^{n}})^{(σ−1)} ∈ μ_{p^{n}}, for all u ∈ E_{L},

and we impose the new condition:

• this map σ coincides with λ_{n} : E_{L} ⟶ E_{L}/μ_{L}E_{K} ⟶^{φ} ℤ ⟶ μ_{p^{n}},

where the left map is the natural one and the right one is surjective (we must choose a primitive p^{n}th root of unity for the right map, but this choice does not change the Frobenius defined up to conjugation: changing the choice of the root of unity is the same as changing the choice of a prime 𝔮′ | 𝔮 in L[μ_{p^{n}}]).

So, the property of φ leads to the following facts:

Let ϕ denotes the map (see the Appendix):

5 See the details in the Appendix.

E_{L} ⟶ R_{G} := {∑_{g} α_{g} g ∈ ℤ[G] | ∑_{g} α_{g} = 0}
u ⟼ ∑_{g} φ(g^{−1}(u)) g ;

Im(ϕ) has finite index, say r. So, p^{δ} being the maximal power of p dividing r, one has:

|{E_{L}/{u ∈ E_{L} | ∀g ∈ G, λ_{n}(g(u)) = 1}| ≥ |R_{G}/R_{G}^{p^{n}}|/p^{δ} = p^{n(|G|−1)−δ} .

On the arithmetical side, the ramification indices in M/L of all primes 𝔮_{L} | 𝔮 of L are all equal to p^{n}; so, with 𝔮_{M}|𝔮_{L}|𝔮 in M/L/K, one has ^{6} :

N_{M/L}(E_{M}) ⊂ N_{M/L}( ∏_{𝔮_{M} |𝔮}U_{𝔮_{M}}) = ∏_{𝔮_{L}|𝔮}U_{𝔮_{L}}^{p^{n}} ,

and then

{u ∈ E_{L} | ∀g ∈ G, λ_{n}(g(u)) = 1} = E_{L} ∩ ∏_{𝔮_{L}|𝔮}U_{𝔮_{L}}^{p^{n}} ⊃ N_{M/L}(E_{M}) ,

so that

|H^{0}(G, E_{M})| = |E_{L}/N_{M/L}(E_{M})| ≥ |E_{L}/{u ∈ E_{L} | ∀g ∈ G, λ_{n}(g(u)) = 1}| ,

and we finally have from the character theory side:

|H^{0}(G, E_{M})| ≥ p^{n(|G|−1)−δ} .

(5) Study of Ĥ^{1}(G, E_{M}) and upper bound for |I_{M}^{G}/P_{M}^{G}|

Recall that M := F L. According to [4], chapter IX, §1, since the cyclic extension M/L splits totally at all infinite places, the Herbrand quotient q(G, E_{M}) of the units is given by:

q(G, E_{M}) := \frac{|Ĥ^{0}(G, E_{M})|}{|Ĥ^{1}(G, E_{M})|} = \frac{1}{[M : L]} = \frac{1}{p^{n+d}} ,

and this gives:

|Ĥ^{1}(G, E_{M})| = p^{n+d} . |Ĥ^{0}(G, E_{M})| ≥ p^{n+d} . p^{n(|G|−1)−δ} = p^{n|G|+d−δ} .

On the other hand, one has the canonical isomorphism:

Ĥ^{1}(G, E_{M}) ≃ P_{M}^{G}/P_{L}

6 Using the fact that the global norm is the product of the corresponding local norms.

where P_{M}^{G} is the group of principal ideals of M which are invariant under G. So one obtains:

|P_{M}^{G}/P_{L}| ≥ p^{n|G|+d−δ}.

Thus, from the formula:^{7}

|I_{M}^{G}/I_{L}| = ∏_{𝔮_{L}|𝔮}e_{𝔮_{L}} = p^{n|G|} ,

where I_{M} is the group of fractional ideals of M, one deduces:

|I_{M}^{G}/P_{M}^{G}| = \frac{|I_{M}^{G}/P_{L}|}{|P_{M}^{G}/P_{L}|} = \frac{|I_{M}^{G}/I_{L}| . |I_{L}/P_{L}|}{|P_{M}^{G}/P_{L}|} = \frac{p^{n|G|} . |Cl_{L}|}{|P_{M}^{G}/P_{L}|} ≤ \frac{p^{n|G|} . |Cl_{L}|}{p^{n|G|+d−δ}} ,

that is:

|I_{M}^{G}/P_{M}^{G}| ≤ |Cl_{L}| p^{δ−d} .

Note that the number in the right hand side does not depend on n.

(6) Does the class c_{L} principalize in M ? Here, we also suppose:

• Frob(𝔮_{L}, H_{L}/L) = c_{L}. M′ being the maximal subfield of M/L in which 𝔮_{L} totally splits, one has:

𝔮_{L} = ∏_{𝔮′_{M} |𝔮_{L} in M′/L}𝔮′_{M} ;

and for all prime 𝔮′_{M}|𝔮_{L} of M′/L, we have 𝔮′_{M} = 𝔮_{M}^{p^{n}}, where 𝔮_{M} is the unique prime of M dividing 𝔮′_{M}. Finally in M,

𝔮_{L} = ( ∏_{𝔮′_{M} |𝔮_{L}} 𝔮_{M} )^{p^{n}} ,

with ∏_{𝔮′_{M} |𝔮_{L}} 𝔮_{M} ∈ I_{M}^{G}. According to Section (5), one has:

|I_{M}^{G}/P_{M}^{G}| ≤ |Cl_{L}| p^{δ−d},

and then:

( ∏_{𝔮′_{M} |𝔮_{L}} 𝔮_{M} )^{|Cl_{L}| . p^{δ−d}} ∈ P_{M}^{G} .

7 Since we have supposed that 𝔮 splits totally in L/K.

Hence, in Cl_{M}, the extended class c_{M} of c_{L} satisfies both:

c_{M} = \overline{𝔮}_{L} = ( ∏_{𝔮′_{M} |𝔮_{L}} \overline{𝔮}_{M} )^{p^{n}} and ( ∏_{𝔮′_{M} |𝔮_{L}} \overline{𝔮}_{M} )^{|Cl_{L}| p^{δ−d}} = 1 .

So, w being such that |Cl_{L}| = p^{w}, c_{L} becomes principal in M under the assumption:

n ≥ w + δ − d.

(7) Existence of 𝔮

We just proved that c_{L} principalizes in M when n is large enough and when 𝔮 (or 𝔮_{L}|𝔮) satisfies the following six conditions:

(1) 𝔮 ∤ p,

(2) μ_{p^{n}} ⊂ K_{𝔮}^{×},

(3) E_{K} ⊂ U_{𝔮}^{p^{n}},

(4) 𝔮 totally splits in L/K,

(5) Frob(𝔮_{L}, L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L) = λ_{n},

(6) Frob(𝔮_{L}, H_{L}/L) = c_{L}.

The definition of λ_{n} ∈ Gal(L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L) shows it is trivial on L[μ_{p^{n}}, E_{K}^{1/p^{n}}]. Then conditions (4) and (5) imply (2) and (3), so we only study compatibility between conditions (1), (4), (5), (6). This compatibility is possible if and only if c_{L} and λ_{n} are equal on the extension H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}] of L.

Let m be the integer such that |(μ_{L})_{p}| = p^{m}; one has, where the exponent ab means Abelian subextension over L:

H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}] ⊂ H_{L} ∩ (L[μ_{p^{n}}, E_{L}^{1/p^{n}}])^{ab} = H_{L} ∩ L[μ_{p^{n+m}}, E_{L}^{1/p^{m}}] ;

let m′ be the integer such that H_{L} ∩ L[μ_{p^{∞}}] = L[μ_{p^{m′}}], so m′ ≥ m and

H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}] = H_{L} ∩ L[μ_{p^{m′}}, E_{L}^{1/p^{m}}] .

The exponent of the Galois group Gal(L[μ_{p^{m′}}, E_{L}^{1/p^{m}}]/L) is less than p^{m′}, so is that of Gal(H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L). According to Section (1), taking a = m′, one can suppose that c_{L} ∈ Cl_{L}^{p^{m′}} (replacing L by L′ as in Section(1); note that m′(L′) = m′(L) because L′/L is unramified at all places dividing p, so that c_{L} ∈ Cl_{L}^{p^{m′(L′)}} as expected); in that case, the restriction of c_{L} is trivial on H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}].

About the restriction of λ_{n} on H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}], we can as well suppose it is trivial, by replacing eventually λ_{n} by λ_{n}^{p^{m′}} (i.e. φ by p^{m′}φ) which has the same properties.^{8}

Up to replacing L by L′ and choosing a convenient φ, the restrictions of c_{L} and of λ_{n} are both trivial on H_{L} ∩ L[μ_{p^{n}}, E_{L}^{1/p^{n}}]: Čebotarev theorem then ensures the existence of infinitely many convenient primes 𝔮 of K satisfying all conditions, and each one gives us an Abelian compositum M in which L principalizes. This proves the Main Theorem.

III. Proofs of the corollaries

Corollary 1 is just the case K = ℚ, and the Kronecker-Weber theorem which states that Abelian extensions of ℚ are cyclotomic.

Corollary 2 is equivalent to the following fact: Let K be a number field and K_{+}^{ab} its maximal ∞-split Abelian extension. Any field L containing K_{+}^{ab} and in which at least one infinite place totally splits over K is principal.

To prove this, let 𝔞 be a fractional ideal of finite type of L. Of course 𝔞 is as well a fractional ideal of a subfield L_{𝔞} of L with finite degree over ℚ. We may assume L_{𝔞} ⊃ K, then L_{𝔞}/K is an extension of number fields in which at least one infinite place is totally split. According to the main theorem, 𝔞 principalizes in an Abelian compositum L_{𝔞}F of L_{𝔞}/K; but L_{𝔞}F is contained in L_{𝔞}K_{+}^{ab} ⊆ L and so 𝔞 becomes principal in L.

Corollary 3 results from corollary 2, taking K = ℚ and K = ℚ(i), respectively.

Corollary 4 is a consequence of Corollary 2.

D – APPENDIX ^{9}

In this Appendix we make precise the construction of the map:

φ : E_{L}/μ_{L}E_{K} ⟶ ℤ,

used by Bosca in Subsection (4). This has some importance since the existence and the properties of φ make use in a crucial manner of the main hypothesis of the paper: “ at least one infinite place of K splits totally in L/K ”; without this assumption the problem of principalization is impossible as explained at the end of the introduction.

8 See the Appendix. 9 Written by G. Gras and J.-F. Jaulent.

The notations are those of the main text.

(8) Definition of φ

Notation: For a group of global units E we denote by \overline{E} the quotient of E by its torsion subgroup μ.

Let N be the norm in L/K and let _{N}E be the kernel of N in E. From the exact sequence:

1 ⟶ _{N}\overline{E}_{L} ⟶ \overline{E}_{L} ⟶ N(\overline{E}_{L}) ⊆ \overline{E}_{K} ⟶ 1 (finite index)

we get:

ℚ ⊗_{ℤ} (\overline{E}_{L}) = ℚ ⊗_{ℤ} (_{N}\overline{E}_{L}) ⊕ ℚ ⊗_{ℤ} (\overline{E}_{K}) i.e. ℚ ⊗_{ℤ} (\overline{E}_{L}/\overline{E}_{K}) = ℚ ⊗_{ℤ} (_{N}\overline{E}_{L}).

Since at least one infinite place of K splits totally in the extension L/K, the Dirichlet–Herbrand theorem implies that the character of ℚ⊗_{ℤ}(_{N}\overline{E}_{L}) contains χ(ℚ[G]) − 1 and that the representation ℚ ⊕ ℚ ⊗_{ℤ} (_{N}\overline{E}_{L}) contains at least a representation R isomorphic to ℚ[G].

We can put R = ℚ ⊗_{ℤ} ⟨ θ ⟩ with θ = ρ . ε_{∗}, where ρ ∈ ℚ^{×}, ρ ≠ ±1, and where ε_{∗} ∈ _{N}\overline{E}_{L} may be seen as a “ relative Minkowski unit ”.

Thus any element u ∈ R is written, in a unique manner, u = θ^{ω} with ω = ∑_{g∈G} α_{g} g ∈ ℚ[G]. It follows that u is a unit (in ⟨ ε_{∗} ⟩_{ℤ[G]}) if and only if ∑_{g∈G} α_{g} = 0. We note that in this case the α_{g} can be taken in \frac{1}{m}ℤ[G] for a suitable m ∈ ℤ (for instance m = | G | in particular cases, but if necessary we can adjust the value of m large enough, multiple of | G |; at the end of the reasoning in Subsection (7), Bosca uses this possibility); for the same reasons, the choice of ε_{∗} is not crucial and ⟨ ε_{∗} ⟩_{ℤ[G]} is not necessarily a direct summand in _{N}\overline{E}_{L}.

In any case m depends only on L/K and not on n.

Noting that

(\overline{E}_{L}/_{N}\overline{E}_{L}) \overline{E}_{K} = \overline{E}_{L}/_{N}\overline{E}_{L} ⊕ \overline{E}_{K}

is killed by | G |, for ε ∈ \overline{E}_{L} we have:

ε^{|G|} = η_{∗} ε_{0}, η_{∗} ∈ _{N}\overline{E}_{L}, ε_{0} ∈ \overline{E}_{K}.

Conclusion. The map φ is defined as follows: Working in ℚ ⊕ ℚ ⊗_{ℤ} (_{N}\overline{E}_{L}), in which R is a direct summand, we associate with ε ∈ \overline{E}_{L} the component of ε^{m} on R, of the form θ^{ω}, with ω = ∑_{g∈G} α_{g} g ∈ ℤ[G], where ∑_{g∈G} α_{g} = 0, then we put:

φ(ε) = α_{1} ∈ ℤ .

This map is trivial on \overline{E}_{K} and defines an element of Hom(E_{L}/μ_{L} . E_{K}, ℤ) with the G-module action defined as usual by:

ψ^{h}(x) := ψ(x^{h^{−1}}), for all ψ ∈ Hom(E_{L}/μ_{L} . E_{K}, ℤ) and all h ∈ G.

It is clear that φ generates a ℤ[G]-submodule whose character is χ(ℤ[G]) − 1. More precisely, a straightforward computation gives φ^{g}(ε) = α_{g} for any g ∈ G, thus ω = ∑_{g∈G} α_{g} g = ∑_{g∈G} φ(ε^{g^{−1}}) g .

In the sequel we put ω =: ϕ(ε).

At this step, Bosca introduces (middle of Subsection (4)) the map λ_{n}:

λ_{n} : E_{L} ⟶ E_{L}/μ_{L} E_{K} ⟶^{φ} ℤ ⟶ μ_{p^{n}} ⟶ 1

by a choice of a primitive p^{n}th root of unity.

This yields an element of Hom(E_{L}, μ_{p^{n}}) which will be by abuse of notation identified, via the Kummer duality between radicals and Galois groups, with the corresponding element σ′ of Gal(L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L[μ_{p^{n}}]).

Then Bosca creates a new condition by saying that σ′ coincide with the Frobenius σ = Frob(𝔮_{L}, L[μ_{p^{n}}, E_{L}^{1/p^{n}}]/L), which is the key idea for the proof of the conjecture. Indeed, the Galois properties of φ give the inequality |H^{0}(G, E_{M})| ≥ p^{n(|G|−1)−δ} (see the last part of Subsection (4)) which is the main ingredient for Subsection (5).

References

[1] Georges Gras, Principalisation d’idéaux par extensions absolument abéliennes, J. Number Th. 62 (1997), 403–421.

[2] Masato Kurihara, On the ideal class group of the maximal real subfields of number fields with all roots of unity, J. European Math. Society 1 (1999), 35–49.

[3] Sébastien Bosca, Capitulations abéliennes, thèse de l’Université Bordeaux 1 (2003).

[4] Serge Lang, Algebraic Number Theory, second edition, Graduate Texts in Mathematics 110 (1994).