A note on free pro-p-extensions of algebraic number fields
Journal de théorie des nombres de Bordeaux, Tome 5 (1993) no. 1, pp. 165-178.

For an algebraic number field k and a prime p, define the number ρ to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro-p-group of rank d. The Leopoldt conjecture implies 1≤ρ≤r 2 +1, (r 2 denotes the number of complex places of k). Some examples of k and p with ρ=r 2 +1 have been known so far. In this note, the invariant ρ is studied, and among other things some examples with ρ<r 2 +1 are given.

Mots clés : algebraic number field, $\mathbb {Z}_p$-extension, free pro-$p$-group 
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Masakazu Yamagishi. A note on free pro-$p$-extensions of algebraic number fields. Journal de théorie des nombres de Bordeaux, Tome 5 (1993) no. 1, pp. 165-178. https://jtnb.centre-mersenne.org/item/JTNB_1993__5_1_165_0/

[1] V.A. Babaicev, 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

[2] G. Gras et J.-F. Jaulent, Sur les corps de nombres réguliers, Math. Z. 202 (1989), 343-365. | MR | Zbl

[3] R. Greenberg, On the structure of certain Galois groups, Invent. Math. 47 (1978), 85-99. | MR | Zbl

[4] K. Iwasawa, On Zl-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. | MR | Zbl

[5] J.-F. Jaulent et T. Nguyen Quang Do, 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

[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. | MR | Zbl

[7] J. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106-132. | MR | Zbl

[8] A. Movahhedi, Sur les p-extensions des corps p-rationnels, Math. Nachr. 149 (1990), 163-176. | MR | Zbl

[9] A. Movahhedi et T. Nguyen Quang Do, Sur l'arithmétique des corps de nombres p-rationnels, Séminaire de Théorie des Nombres, Paris 1987-88, Progr. Math., 81, Birkhäuser Boston, MA,1990, 155-200. | MR | Zbl

[10] J. Neukirch, Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Archiv der Math. 22 (1971), 337-357. | MR | Zbl

[11] T. Nguyen Quang Do, Sur la structure galoisienne des corps locaux et la théorie d'Iwasawa, Compositio Math. 46 (1982), 85-119. | Numdam | MR | Zbl

[12] T. Nguyen Quang Do, Formations de classes et modules d'Iwasawa, Number Theory Noordwijkerhout 1983, Lecture Notes in Math. 1068 (1984), 167-185. | MR | Zbl

[13] T. Nguyen Quang Do, Sur la torsion de certains modules galoisiens II, Séminaire de Théorie des Nombres, Paris 1986-87, Progr. Math., 75, Birkhäuser Boston, MA, 1988, 271-297. | MR | Zbl

[14] I.R. Šafarevic, 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. | Numdam

[15] J.P. Serre, Cohomologie galoisienne, Lecture Notes in Math. 5 (1964). | MR | Zbl

[16] J. Sonn, Epimorphisms of Demushkin groups, Israel J. Math. 17 (1974), 176-190. | MR | Zbl

[17] V.M. Tsvetkov, 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. | MR | Zbl

[18] K. Wingberg, Freie Produktzerlegungen von Galoisgruppen und Iwasawa-Invarianten für p-Erweiterungen von Q, J. Reine Angew. Math. 341 (1983), 111-129. | MR | Zbl

[19] K. Wingberg, Duality theorems for Γ-extensions of algebraic number fields, Compositio Math. 55 (1985), 333-381. | Numdam | Zbl

[20] K. Wingberg, On Galois groups of p-closed algebraic number fields with restricted ramification, J. Reine Angew. Math. 400 (1989), 185-202. | MR | Zbl

[21] K. Wingberg, On Galois groups of p-closed algebraic number fields with restricted ramification II, J. Reine Angew. Math. 416 (1991), 187-194. | MR | Zbl

[22] M. Yamagishi, 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. | MR | Zbl