Note on 2-rational fields

Georges Gras & Jean-François Jaulent

Résumé. Nous déterminons le groupe de Galois de la pro-2-extension 2-ramifiée maximale d’un corps de nombres 2-rationnel.

Abstract. We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field.

Nota. This short Note is motivated by the paper “Galois 2-extensions unramified outside 2” of J. Jossey and, at this occasion, we bring into focus some classical technics of abelian ℓ-ramification which, unfortunately, are often ignored, especially those developped by J-F. Jaulent with the ℓ-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify proofs in such subjects; for instance, the main Theorem 2, due to J-F. Jaulent, generalizes the purpose of Jossey’s paper in such a way.

1 Introduction and history

The notions of ℓ-rational field and ℓ-regular field (for a prime number ℓ and a number field K), independently introduced by A. Movahhedi and T. Nguyen Quang Do in [MN], and by G. Gras and J-F. Jaulent in [GJ], coincide as soon as K contains the maximal real subfield of the field of ℓth roots of unity, thus especially for ℓ = 2.

• The ℓ-regularity expresses the triviality of the regular ℓ-kernel of K (i.e. the kernel, in the ℓ-part of the universal group K_{2}(K), of Hilbert symbols attached to the non-complex places not dividing ℓ).

• The ℓ-rationality traduces the pro-ℓ-freeness of the Galois group G_{K} := Gal(M_{K}/K) of the maximal pro-ℓ-extension ℓ-ramified ∞-split M_{K} of K (i.e. unramified at the finite places^{1} not dividing ℓ and totally split at the infinite places).

More precisely, let c_{K} be the number of complex places of K; let μ_{K} (resp. μ_{K_{𝔩}}) be the ℓ-group of roots of unity in K (resp. in the localization K_{𝔩}); and let

V_{K} := {x ∈ K^{×}| x ∈ K_{𝔩}^{×ℓ} ∀𝔩 | ℓ & v_{𝔭}(x) ≡ 0 mod ℓ ∀𝔭 ∤ ℓ∞} be the group of ℓ-hyperprimary elements in K^{×}. Then, with these notations, from [JN, Th.1.2] or [G_{3}, IV.3.5, III.4.2.3], the ℓ-rationality of K may be expressed as follows:

1According to the conventions of the ℓ-adic class field theory (cf. [G_{3}, Ja]), we never speak of ramification at infinity but of complexification of real places.

Theorem and definition 0. The following conditions are equivalent:

(i) The Galois group G_{K} is a free pro-ℓ-group on 1 + c_{K} generators. (ii) The abelianization G_{K}^{ab} of G_{K} is a free ℤ_{ℓ}-module of dimension 1+c_{K}. (iii) The field K satisfies the Leopoldt conjecture (for the prime ℓ) and the torsion submodule T_{K} of G_{K}^{ab} is trivial. (iv) One has the equalities: V_{K} = K^{×ℓ} & rk_{ℓ} (μ_{K}) = ∑_{𝔩|ℓ} rk_{ℓ} (μ_{K_{𝔩}}).

When any of these conditions is realized, the number field K is said to be ℓ-rational.

Remark. In case ℓ = 2, it follows from the condition (iv) above that a 2rational field has a single place above 2.

The premises of the notion of ℓ-regularity go back to the works of G. Gras, mainly to his note on the K_{2} of number fields [G_{2}, II, § 2; III, §§ 1, 2], whereas the notion of ℓ-rationality appears (in a hidden form) in the work of H. Miki [Mi] concerning the study of a sufficient condition for the Leopoldt conjecture, as well as those of K. Wingberg [W_{1}, W_{2}], concerning the same condition.

Movahhedi’s thesis and the above papers [GJ, MN] characterised the going up for ℓ-rationality in any ℓ-extension in terms of ℓ- primitivity of the ramification (a definition given in [G_{2}, III, § 1] from the use of the Log function defined in [G_{1}]), a property which was unknown in the preceding approaches.

For instance, this gives immediately that if K is a ℓ-extension of ℚ, a N.S.C. for K to be ℓ-rational is that K/ℚ be ℓ-ramified, or that K/ℚ be {p, ℓ}-ramified, where p ≠ ℓ is a prime ≡ 1 mod (ℓ) such that p ≢ ±1 mod (8) if ℓ = 2 and p ≢ 1 mod (ℓ^{2}) if ℓ ≠ 2 (cf. [G_{3}, IV.3.5.1] giving Jossey’s examples [Jo]).

A synthesis of these results is given in [JN] and a systematic exposition is developped in the book of G. Gras ([G_{3}, III, § 4, (b); IV, § 3, (b); App., § 2]).

Various generalizations of these notions have been studied by O. Sauzet and J-F. Jaulent (cf. [JS_{1}, JS_{2}]), especially in the case ℓ = 2 which is, as usual, the most triky; in particular, they introduce the notion of 2-birational fields.

Very recently, J. Jossey [Jo] introduced a notion of ℓ-rationality which is incompatible with the classical one (for ℓ = 2, as soon as K contains real embeddings) and is unlucky since it does not apply to the field of rationals ℚ.

For these reasons, to avoid any confusion, we propose to speak, in his context, of 2-superrational fields. More precisely:

Definition 1. Let K be a number field with r_{K} real places and c_{K} complex places, M′_{K} the maximal 2-ramified pro-2-extension of K, and M_{K} the maximal subextension of M′_{K} totally split at the infinite places. We say that K is:

(i) 2-superrational, if G′_{K} := Gal(M′_{K}/K) is pro-2-free;

(ii) 2-rational, if its quotient G_{K} := Gal(M_{K}/K) is pro-2-free.

The purpose of the next section is to determine the structure of the Galois group G′_{K} when the number field K is 2-rational.

Our proof relies on the functorial properties of ℓ-ramification theory.

2 Main Theorem: description of G′_{K} = Gal(M′_{K}/K)

Our very simple result has the following statement:

Theorem 2. Let K be a 2-rational number field having r_{K} real places and c_{K} complex places. The Galois group G′_{K} := Gal(M′_{K}/K) of the maximal 2-ramified pro-2-extension M′_{K} of K is the pro-2-free product

G′_{K} ≃ ℤ_{2}^{⊛(1+c_{K})} ⊛ (ℤ/2ℤ)^{⊛ r_{K}}

of (1 + c_{K}) copies of the procyclic group ℤ_{2} and of r_{K} copies of ℤ/2ℤ.

Corollary 3. The 2-rational number fields which are 2-superrational are the totally imaginary ones.

Proof. Consider the quadratic extension L = K[i] generated by the 4th roots of unity. It is 2-ramified over K, thus thanks to the going up theorem of [GJ, MN] (cf. e.g. [JN, Th. 3.5] or [G_{3}, IV.3.4.3, (iii)]), it is 2-rational, then 2superrational since it is totally imaginary. In other words, the Galois group G_{L} = G′_{L} of the maximal 2-ramified pro-2-extension M_{L} of L is pro-2-free.

Since the quadratic extension L/K is 2-ramified, M_{L} is also the maximal 2-ramified pro-2-extension M′_{K} of K; the Galois group G′_{K} is potentially free since it contains the pro-2-free open subgroup G_{L} of index 2 in G′_{K}.

As in [Jo], the results of W. Herfort and P. Zalesskii (cf. [HZ, Th.0.2]) give the existence of a finite familly (F_{i})_{i=0,...,k} of free pro-2-groups on respectively d_{0}, …, d_{k} generators (where k is the number of conjugacy classes of subgroups of order 2 in G′_{K}), such that:

G′_{K} ≃ F_{0} ⊛ ( ⊛_{i=1}^{k}(F_{i} × ℤ/2ℤ)).

In particular, the abelianisation G_{K}^{′ab} of G′_{K} admits the direct decomposition:

G_{K}^{′ab} ≃ ℤ_{2}^{d_{0}} ⊕ (⊕_{i=1}^{k}(ℤ_{2}^{d_{i}} ⊕ ℤ/2ℤ)) ≃ ℤ_{2}^{d_{0}+d_{1}+···+d_{k}} ⊕ (ℤ/2ℤ)^{k}.

Since the 2-rational field K satisfies the Leopoldt conjecture, we get ∑_{i=0}^{k} d_{i} = 1 + c_{K} as well as the isomorphism T′_{K} := tor_{ℤ_{2}}(G_{K}^{′ab}) ≃ (ℤ/2ℤ)^{k}. Moreover T_{K} := tor_{ℤ_{2}}(G_{K}^{ab}) = 1, so that T′_{K} is generated by the decomposition groups of the real places of K which are deployed, a key argument of class field theory (cf. [Ja] or [G_{3}, III.4.1.5]) giving k = r_{K}.

Now the pro-2-decomposition of G′_{K} clearly shows that the minimal number of generators d(G′_{K}) and of relations r(G′_{K}), defining G′_{K} as a pro-2-group, are:

d(G′_{K}) = k + ∑_{i=0}^{k} d_{i} = r_{K} + 1 + c_{K} and r(G′_{K}) = ∑_{i=1}^{k}(1 + d_{i}) = d(G′_{K}) − d_{0}.

It is well-known (cf. e.g. [G_{3}, App., Th.2.2, (i)]) that one has^{2}:

d(G′_{K}) − r(G′_{K}) = dim_{𝔽_{2}}(H^{1}(G′_{K}, 𝔽_{2})) − dim_{𝔽_{2}}(H^{2}(G′_{K}, 𝔽_{2})) = 1 + c_{K}.

Thus we obtain d_{0} = 1 + c_{K}, giving d_{i} = 0 for 1 ≤ i ≤ k, then the expected result.

2This argument is equivalent to the use of the formulas of Šafarevič (cf. [Sa] or [NSW, Th.8.7.3])

References

[G_{1}] G. Gras, Logarithme p-adique et groupes de Galois, J. reine angew. Math. 343

(1983), 64–80. [G_{2}] G. Gras, Remarks on K_{2} of number fields, J. Number Th. 23 (1986), 64–80 . [G_{3}] G. Gras, Class Field Theory : from theory to practice, SMM 2003 ; second

corrected printing 2005.

[GJ] G. Gras & J-F. Jaulent, Sur les corps de nombres réguliers, Math. Z. 202 (1989), 343–365.

[HZ] W. Herfort & P. Zalesskii, Finitely generated pro-2-groups with a free subgroup of index 2, Manuscripta Math. 93 (1997), 457–464.

[Ja] J-F. Jaulent, Théorie ℓ-adique globale du corps de classes, J. Théor. Nombres Bordeaux 10 (1998), 355–397.

[JN] J-F. Jaulent & T. Nguyen Quang Do, Corps p-rationnels, corps p-réguliers et ramification restreinte, J. Théor. Nombres Bordeaux 5 (1993), 343–365.

[JS_{1}] J-F. Jaulent & O. Sauzet, Pro-ℓ-extensions de corps de nombres 𝔩-rationnels,

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totalement réels, Pub. Matemàtiques 44 (2000), 343–351.

[Jo] J. Jossey Galois 2-extensions unramified outside 2, J. Number Th. 124 (2007), 42–76.

[Mi] H. Miki, On the Leopoldt conjecture on the p-adic regulators, J. Numb. Th. 26 (1987), 117–128.

[MN] A. Movahhedi & T. Nguyen Quang Do, Sur l’arithmétique des corps de nombres p-rationnels, Sém. Th. Nombres Paris (1987/1988), Prog. in Math. 89 (1990), 155–200.

[NSW] J. Neukirch, A. Schmidt & K. Wingberg, Cohomology of number fields, Grund. der Math. Wissenschaften 323, Springer-Verlag 2000 ; second edition 2008.

[Sa] I. R. Šafarevič, Extensions with prescribed ramification points, Pub. Math. I.H.E.S. 36 (1986), 71–95.

[W_{1}] K. Wingberg, On Galois groups of p-closed algebraic number fields with restricted ramification, J. reine angew. Math. 400 (1989), 185–202.

[W_{2}] K. Wingberg, On Galois groups of p-closed algebraic number fields with restricted ramification II, J. reine angew. Math. 416 (1991), 187–194.

Georges Gras  Jean-François Jaulent

Institut de Mathématiques

Villa la Gardette  Université de Bordeaux
chemin Château Gagnière  351, cours de la Libération
F-38520 Le Bourg d’Oisans  F-33405 TALENCE Cedex
g.mn.gras@wanadoo.fr  jaulent@math.u-bordeaux1.fr