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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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