×

A note on free pro-\(p\)-extensions of algebraic number fields. (English) Zbl 0784.11052

The author studies the maximal rank \(\rho\) of free pro-\(p\) Galois groups \(F_ \rho\) over a fixed algebraic number field \(k\) for a fixed prime \(p\). Such an \(F_ \rho\) is necessarily a quotient of \(G_{S_ p}\), the Galois group over \(k\) of the maximal pro-\(p\)-extension of \(k\) unramified outside \(p\). By class field theory, the rank of \(G_{S_ p}^{ab}\) is \(1+r_ 2+\delta\), where \(r_ 2\) denotes the number of complex places of \(k\), and \(\delta\) the defect of Leopoldt’s conjecture. The main results are the following:
1) \(\rho\leq 1+r_ 2\) if the “weak Leopoldt conjecture” holds for all \(\mathbb{Z}_ p\)-extensions of \(k\).
2) There exist \(k\), \(p\) such that \(\rho<1+r_ 2\). Such examples come from number fields \(k\) for which \(G_{S_ p}\), under rather strong conditions, is a quotient of a free pro-\(p\)-product of decomposition groups and a free group [see K. Wingberg, J. Reine Angew. Math. 400, 185-202 (1989; Zbl 0715.11065)].

MSC:

11R32 Galois theory
11R37 Class field theory
11R34 Galois cohomology

Citations:

Zbl 0715.11065
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Babaicev, V.A., On some questions in the theory of Γ-extensions of algebraic number fields, Izv. Akad. Nauk. SSSR. Ser. Mat.40 (1976), 477-487; English transl. in Math. USSR-Izv.10 (1976), 453-462. · Zbl 0366.12005
[2] Gras, G. et Jaulent, J.-F., Sur les corps de nombres réguliers, Math. Z.202 (1989), 343-365. · Zbl 0704.11040
[3] Greenberg, R., On the structure of certain Galois groups, Invent. Math.47 (1978), 85-99. · Zbl 0403.12004
[4] Iwasawa, K., On Zl-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. · Zbl 0285.12008
[5] Jaulent, J.-F. et Nguyen Quang Do, T., Corps p-rationnels, corps p-réguliers, et ramification restreinte, Séminaire de Théorie des Nombres de Bordeaux, (1987-1988), Exposé 10, 10-01-10-26. · Zbl 0748.11052
[6] L, V.Kuz’min, Local extensions associated with l-extensions with given ramification, Izv. Akad. Nauk. SSSR. Ser. Mat.39 (1975), 739-772; English transl. in Math. USSR-Izv.9 (1975), 693-726. · Zbl 0342.12007
[7] Labute, J., Classification of Demushkin groups, Canad. J. Math.19 (1967), 106-132. · Zbl 0153.04202
[8] Movahhedi, A., Sur les p-extensions des corps p-rationnels, Math. Nachr.149 (1990), 163-176. · Zbl 0723.11054
[9] Movahhedi, A. et Nguyen Quang Do, T., Sur l’arithmétique des corps de nombres p-rationnels, Séminaire de Théorie des Nombres, Paris1987-88, Progr. Math., 81, BirkhäuserBoston, MA,1990, 155-200. · Zbl 0703.11059
[10] Neukirch, J., Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Archiv der Math.22 (1971), 337-357. · Zbl 0254.20023
[11] Nguyen Quang Do, T., Sur la structure galoisienne des corps locaux et la théorie d’Iwasawa, Compositio Math.46 (1982), 85-119. · Zbl 0481.12004
[12] Nguyen Quang Do, T., Formations de classes et modules d’Iwasawa, Number Theory Noordwijkerhout1983, 1068 (1984), 167-185. · Zbl 0543.12007
[13] Nguyen Quang Do, T., Sur la torsion de certains modules galoisiens II, Séminaire de Théorie des Nombres, Paris1986-87, Progr. Math., 75, BirkhäuserBoston, MA, 1988, 271-297. · Zbl 0687.12005
[14] Šafarevic, I.R., Extensions with given points of ramification, Inst. Hautes Études Sci. Publ. Math.18 (1964), 295-319; English transl. in Amer. Math. Soc. Transl. Ser. 2 59 (1966), 128-149; see also Collected Mathematical Papers, 295-316. · Zbl 0118.27505
[15] Serre, J.P., Cohomologie galoisienne, 5 (1964). · Zbl 0812.12002
[16] Sonn, J., Epimorphisms of Demushkin groups, Israel J. Math.17 (1974), 176-190. · Zbl 0286.12010
[17] Tsvetkov, V.M., Examples of extensions with Demushkin group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 103 (1980), 146-149; English transl. in J. Soviet Math.24-4 (1984), 480-482. · Zbl 0472.12008
[18] Wingberg, K., Freie Produktzerlegungen von Galoisgruppen und Iwasawa-Invarianten für p-Erweiterungen von Q, J. Reine Angew. Math.341 (1983), 111-129. · Zbl 0501.12014
[19] Wingberg, K., Duality theorems for Γ-extensions of algebraic number fields, Compositio Math.55 (1985), 333-381. · Zbl 0608.12012
[20] Wingberg, K., On Galois groups of p-closed algebraic number fields with restricted ramification, J. Reine Angew. Math.400 (1989), 185-202. · Zbl 0715.11065
[21] Wingberg, K., On Galois groups of p-closed algebraic number fields with restricted ramification II, J. Reine Angew. Math.416 (1991), 187-194. · Zbl 0728.11058
[22] Yamagishi, M., On the center of Galois groups of maximal pro-p extensions of algebraic number fields with restricted ramification, J. Reine Angew. Math.436 (1993), 197-208. · Zbl 0766.11044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.