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On the towers of torsion Bertrandias and Payan modules. (English) Zbl 1430.11150
Summary: For an odd prime $$p$$, let $$K/k$$ be a Galois $$p$$-extension and $$S$$ be a set of primes of $$k$$ containing the primes lying over $$p$$. For the $$p^r$$th roots $${\mu _{{p^r}}}(K)$$ of unity in $$K$$, we describe the so-called Sha group $$\mathrm{Sha}_{S}(G(K/k), {\mu _{{p^r}}}(K))$$ in terms of the Galois groups of certain subfields of $$K$$ corresponding to $$S$$. As an application, we investigate a tower of extension fields $$\{k_{T^i}\}_{i \geq 0}$$ where $$k_{T^{i + 1}}$$ is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of $$k_{T^i}$$ over $$k_{T^i}$$. This is called a tower of torsion parts of the Bertrandias and Payan extensions over $$k$$. We find a relation between the degrees $$\{[k_{{T^{i + 1}}}:k_{{T^i}}]\}_{i \geq 0}$$ over the towers. Using this formula we investigate whether the towers are stationary or not.
MSC:
 11R23 Iwasawa theory 11R34 Galois cohomology
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References:
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