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A finiteness theorem in the Galois cohomology of algebraic number fields. (English) Zbl 0648.12009

Soit \(k\) un corps de nombres, \(\mathfrak g= \mathrm{Gal}(\bar k/k)\) et \(M\) un \(\mathbb{Z}_{\ell}\)-module de type fini muni d’une action continue de \(\mathfrak g\). Si \(M\) est non ramifié presque partout, et si pour les places non ramifiées non au-dessus de \(\ell\), l’élément de Frobenius n’a de points of higher order and certain sequences of orderings of higher level called “chains”.
In the paper under review the author studies these chains and shows that this study can be reduced to tractable problems in abelian group theory and in the theory of real closures of level 1, i.e., the real closed fields introduced by Artin and Schreier. Moreover, the author’s methods enable him to study chains under field extensions, subfield formation, and places.
There is a paper by N. Schwartz dealing with related questions [J. Algebra 110, 74–107 (1987; Zbl 0632.12022)] which, it should be noted, was submitted over three years after the paper under review was.

MSC:

11R34 Galois cohomology
14K15 Arithmetic ground fields for abelian varieties
14G25 Global ground fields in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies

Citations:

Zbl 0632.12022
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References:

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