On the towers of torsion Bertrandias and Payan modules

Abstract

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 $${\mu _{{p^r}}}\left( K \right)$$ of unity in K, we describe the so-called Sha group Sha S (G(K/k), $${\mu _{{p^r}}}\left( K \right)$$ ) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields $${\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0$$ 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 $${\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}$$ over the towers. Using this formula we investigate whether the towers are stationary or not.