On regular number fields. (Sur les corps de nombres réguliers.) (French) Zbl 0704.11040

Fix a prime number \(\ell\) and let \(\zeta\) denote a primitive \(\ell\)-th root of unity. Let \(K\) be a number field with divisor group \(D_ K\) and write \(\mathcal D_ K\) for the multiplicative tensor product \(\mathbb Z_{\ell}\otimes_{\mathbb Z}D_ K\). One can define a surjective map (called Gras’s logarithm), \(\text{lg}: \mathcal D_ K\to \text{Gal}(Z/K)\), where \(Z\) is the composite field of the \(\mathbb Z_{\ell}\)-extensions of \(K\). For a finite or a real place \(\wp\) of \(K\), \(\text{lg}(\wp)=1\) whenever \(\wp\) is real or lies over \(\ell\), otherwise \(\text{lg}(\wp)\) is a topological generator of the decomposition group \(D_{\wp}(Z/K)\simeq\mathbb Z_{\ell}\) associated to \(\wp\) in the abelian extension \(Z/K\). A finite set \(S\) of places of \(K\) is called primitive when the \(\text{lg}(s)\), \(s\in S\), form a \(\mathbb Z_{\ell}\)-basis of a pure submodule of \(\text{Gal}(Z/K)\). An \(\ell\)-extension \(L/K\) is called primitively ramified if the set \(S\) of places of \(K\) that ramify tamely in \(L/K\), is primitive. \(K\) is called regular (with respect to \(\ell\)) if the \(\ell\)-Sylow subgroup \(R_ 2(K)\) of the kernel in \(K_ 2(K)\) of the regular symbols attached to the non-complex places of \(K\), is trivial. When \(K\) contains the maximal real subfield \(k=\mathbb Q(\zeta +\zeta^{-1})\) of the cyclotomic field \(\mathbb Q(\zeta)\) several equivalent characterizations of regularity can be given, one of which says that \(K\) is regular if and only if \(K\) verifies Leopoldt’s conjecture (with respect to \(\ell)\) and the torsion submodule \({\mathcal T}_ K\) of \(\text{Gal}(M/K)\), with \(M\) the maximal \(\ell\)-ramified, abelian \(\ell\)-extension of \(K\) which decomposes completely at the infinite places, is zero. A classical example of a regular number field is provided by the cyclotomic field \(\mathbb Q(\zeta)\) if and only if \(\ell\) is a regular prime in the usual terminology. Writing \(\delta_ K\) for the defect of Leopoldt’s conjecture for \(K\), \(K\) is called an \(\ell\)-rational field if \({\mathcal T}_ K=\{0\}\) and \(\delta_ K=0\). The main results of the paper can now be formulated:
1) Let \(K\) contain the maximal real subfield \(k\), and let \(L/K\) be a Galois \(\ell\)-extension. Then the following conditions are equivalent: (i) \(L\) is regular; (ii) \(K\) is regular and \(L/K\) is primitively ramified.
2) Let \(L/K\) be a Galois \(\ell\)-extension. Then the following conditions are equivalent: (i) \(L\) is \(\ell\)-rational; (ii) \(K\) is \(\ell\)-rational and \(L/K\) is primitively ramified.


11R29 Class numbers, class groups, discriminants
11R70 \(K\)-theory of global fields
11S15 Ramification and extension theory
19C99 Steinberg groups and \(K_2\)
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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[1] Conner, P.E., Hurrelbrink, J.: A comparison theorem for the 2-rank ofK 2(O). Contemp. Math.55, 411-420 (1986) · Zbl 0598.12012
[2] Gras, G.: Groupe de Galois de lap-extension ab?liennep-ramifi?e maximale d’un corps de nombres. J. Reine Angew. Math.333, 86-132 (1982) · Zbl 0477.12009
[3] Gras, G.: Logarithmep-adique et groupes de Galois. J. Reine Angew. Math.343, 64-80 (1983) · Zbl 0501.12015
[4] Gras, G.: Decomposition and Inertia groups in? p -extensions. J. Fac. Sci., Univ. Tokyo, Sect. 1A9, 41-51 (1986) · Zbl 0606.12003
[5] Gras, G.: Remarks onK 2 of number fields. J. Number Theory23, 322-335 (1986) · Zbl 0589.12010
[6] Jaulent, J.-F.: Introduction auK 2 des corps de nombres. Publ. Math. Fac. Sci. Besan?on, Th?or. Nombres 1981/1982, 1982/1983 (1983)
[7] Jaulent, J.-F.: L’arithm?tique des ?-extensions (Th?se). Publ. Math. Fac. Sci. Besan?on, Th?or. Nombres 1984/1985, 1985/1986 (1986)
[8] Kubota, T.: Galois group of the abelian maximal extension of an algebraic number field. Nagoya Math. J.12, 177-189 (1957) · Zbl 0079.26803
[9] Miki, H.: On the Leopoldt conjecture on thep-adic regulators. J. Number Theory26, 117-128 (1987) · Zbl 0621.12009
[10] Miki, H., Sato, H.: Leopoldt’s conjecture and Reiner’s theorem. J. Math. Soc. Japan36, 47-52 (1984) · Zbl 0534.12005
[11] Movahhedi, A.: Sur lesp-extensions des corpsp-rationnels (Th?se) (1988)
[12] Nguyen Quang Do, T.: Miki’s theorem revisited (Pr?publication)
[13] Serre, J.-P.: Corps locaux. Paris: Hermann 1968
[14] Tate, J.: Relations betweenK 2 and Galois cohomology. Invent. Math.36, 257-274 (1976) · Zbl 0359.12011
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