### 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 μpr(K) of unity in K, we describe the so-called Sha group Sha_{S}(G(K/k), μpr(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 kTii≥0 where kTi+1 is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of kTi over kTi. This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees {[kTi+1:kTi]}i≥0 over the towers. Using this formula we investigate whether the towers are stationary or not.

Original language | English |
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Pages (from-to) | 563-583 |

Number of pages | 21 |

Journal | Israel Journal of Mathematics |

Volume | 221 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 Sep 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)