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Galois groups of number fields generated by torsion points of elliptic curves. (English) Zbl 0621.12011
Let $$F$$ be an algebraic number field. Let $$p\neq 2$$ be a prime, $$S_ p$$ the set of all primes in $$F$$ above $$p$$, $$S$$ a finite set of primes in $$F$$, $$F_ S$$ the maximal p-extension of $$F$$ unramified outside $$S$$, and $$F_{\infty}^ a$$ $${\mathbb{Z}}_ p$$-extension in $$F_ S$$. The main result in this article is a general theorem giving sufficient conditions for the Galois groups $$G(F_{S_ p}/F_{\infty})$$, $$G(F_{S\cap S_ p}/F_{\infty})$$, $$G(F_ S/F_{\infty})$$, and $$G(F_{S\cup S_ p}/F_{\infty})$$ to be free pro-p-groups. If $$T\supseteq S$$, sufficient conditions are given for $$G(F_ T/F_ S)$$ to be a free pro-p-product of certain inertia groups. This generalizes results of J. Neukirch [J. Reine Angew. Math. 259, 1-47 (1973; Zbl 0263.12006)] and O. Neumann [in Algebr. Number Fields, Proc. Symp. London math. Soc., Univ. Durham 1975, 625-647 (1977; Zbl 0393.12015)].
Reviewer: L.Olson

##### MSC:
 11R32 Galois theory 11R18 Cyclotomic extensions 14H52 Elliptic curves 11R34 Galois cohomology 11S15 Ramification and extension theory 14G25 Global ground fields in algebraic geometry
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##### References:
 [1] Composite Math 55 pp 333– (1985) [2] On p-closed number fields and an analogue of Riemann’s existence theorem pp 625– (1977) [3] J. reine angew. Math 259 pp 1– (1973) [4] DOI: 10.1007/BF01393198 · Zbl 0534.12009 · doi:10.1007/BF01393198 [5] DOI: 10.1007/BF01388840 · Zbl 0563.12008 · doi:10.1007/BF01388840 [6] Galois cohomology of algebraic number fields (1978) [7] Progress in Mathematics Vol. 35 (Birkhäuser) Arithmetic and Geometry I dedicated to I.R (1983) [8] DOI: 10.1007/BF01402975 · Zbl 0359.14009 · doi:10.1007/BF01402975 [9] DOI: 10.2307/1970784 · Zbl 0285.12008 · doi:10.2307/1970784
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