For an algebraic number field and a prime , define the number to be the maximal number such that there exists a Galois extension of whose Galois group is a free pro--group of rank . The Leopoldt conjecture implies , ( denotes the number of complex places of ). Some examples of and with have been known so far. In this note, the invariant is studied, and among other things some examples with are given.
@article{JTNB_1993__5_1_165_0, author = {Yamagishi, Masakazu}, title = {A note on free pro-$p$-extensions of algebraic number fields}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {165--178}, publisher = {Universit\'e Bordeaux I}, volume = {5}, number = {1}, year = {1993}, zbl = {0784.11052}, mrnumber = {1251235}, language = {en}, url = {http://www.numdam.org/item/JTNB_1993__5_1_165_0/} }
TY - JOUR AU - Yamagishi, Masakazu TI - A note on free pro-$p$-extensions of algebraic number fields JO - Journal de Théorie des Nombres de Bordeaux PY - 1993 DA - 1993/// SP - 165 EP - 178 VL - 5 IS - 1 PB - Université Bordeaux I UR - http://www.numdam.org/item/JTNB_1993__5_1_165_0/ UR - https://zbmath.org/?q=an%3A0784.11052 UR - https://www.ams.org/mathscinet-getitem?mr=1251235 LA - en ID - JTNB_1993__5_1_165_0 ER -
Yamagishi, Masakazu. 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. http://www.numdam.org/item/JTNB_1993__5_1_165_0/
[1] 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] Sur les corps de nombres réguliers, Math. Z. 202 (1989), 343-365. | MR 1017575 | Zbl 0704.11040
et ,[3] On the structure of certain Galois groups, Invent. Math. 47 (1978), 85-99. | MR 504453 | Zbl 0403.12004
,[4] On Zl-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. | MR 349627 | Zbl 0285.12008
,[5] 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
et ,[6] 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 392925 | Zbl 0342.12007
[7] Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106-132. | MR 210788 | Zbl 0153.04202
,[8] Sur les p-extensions des corps p-rationnels, Math. Nachr. 149 (1990), 163-176. | MR 1124802 | Zbl 0723.11054
,[9] 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 1042770 | Zbl 0703.11059
et ,[10] Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Archiv der Math. 22 (1971), 337-357. | MR 347992 | Zbl 0254.20023
,[11] Sur la structure galoisienne des corps locaux et la théorie d'Iwasawa, Compositio Math. 46 (1982), 85-119. | Numdam | MR 660155 | Zbl 0481.12004
,[12] Formations de classes et modules d'Iwasawa, Number Theory Noordwijkerhout 1983, Lecture Notes in Math. 1068 (1984), 167-185. | MR 756093 | Zbl 0543.12007
,[13] 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 990514 | Zbl 0687.12005
,[14] 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] Cohomologie galoisienne, Lecture Notes in Math. 5 (1964). | MR 201444 | Zbl 0812.12002
,[16] Epimorphisms of Demushkin groups, Israel J. Math. 17 (1974), 176-190. | MR 349636 | Zbl 0286.12010
,[17] 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 618509 | Zbl 0472.12008
,[18] Freie Produktzerlegungen von Galoisgruppen und Iwasawa-Invarianten für p-Erweiterungen von Q, J. Reine Angew. Math. 341 (1983), 111-129. | MR 697311 | Zbl 0501.12014
,[19] Duality theorems for Γ-extensions of algebraic number fields, Compositio Math. 55 (1985), 333-381. | Numdam | Zbl 0608.12012
,[20] On Galois groups of p-closed algebraic number fields with restricted ramification, J. Reine Angew. Math. 400 (1989), 185-202. | MR 1013730 | Zbl 0715.11065
,[21] On Galois groups of p-closed algebraic number fields with restricted ramification II, J. Reine Angew. Math. 416 (1991), 187-194. | MR 1099949 | Zbl 0728.11058
,[22] 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 1207286 | Zbl 0766.11044
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