an:06808410
Zbl 1430.11150
Seo, Soogil
On the towers of torsion Bertrandias and Payan modules
EN
Isr. J. Math. 221, No. 2, 563-583 (2017).
00370992
2017
j
11R23 11R34
Bertrandias and Payan modules; torsion; Galois groups
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.