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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. Akad. Nauk SSSR Ser. Mat., 1975 Volume 39, Issue 3, Pages 487–495 (Mi im2038)

This article is cited in 6 papers

Cohomological dimension of some Galois groups

L. V. Kuz'min


Abstract: Suppose that $l$ is a prime number, $k$ is an algebraic number field containing a primitive root $\zeta_l$ ($\zeta_4$ if $l=2$), $S$ is a finite set of places of $k$ which contains all divisors of $l$, $K$ is the maximal $l$-extension of $k$ unramified outside $S$, $k_\infty$ is an arbitrary $\Gamma$-extension of $k$, and $H=G(K/k_\infty$. In this paper we find necessary and sufficient conditions for the group $H$ to be a free pro-$l$-group. We also obtain a description of all $\Gamma$-extensions $k_\infty/k$ having the property that any place of $k$ has a finite number of extensions to $k_\infty$. We prove that, in some sense, such $\Gamma$-extensions make up the overwhelming majority of all $\Gamma$-extensions.
Bibliography: 4 items.

UDC: 519.4

MSC: Primary 12A60; Secondary 12G10, 12A55, 12A65, 12F10

Received: 18.06.1974


 English version:
Mathematics of the USSR-Izvestiya, 1975, 9:3, 455–463

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