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On the Bertrandias-Payan module in a $$p$$-extension – capitulation kernel. (Sur le module de Bertrandias-Payan dans une $$p$$-extension – Noyau de capitulation.) (French. English summary) Zbl 1406.11112
Algèbre et théorie des nombres 2016. Besançon: Presses Universitaires de Franche-Comté. Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2016, 25-44 (2017).
Summary: For a number field $$K$$ and a prime number $$p$$ we denote by $$BP_K$$ the compositum of the cyclic $$p$$-extensions of $$K$$ which are embeddable into a cyclic $$p$$-extension of arbitrary large degree. The extension $$BP_K/K$$ is $$p$$-ramified and is a finite extension of the compositum $$\tilde{K}$$ of the $$\mathbb Z_p$$-extensions of $$K$$. The group $$\mathcal{BP}_K := \mathrm{Gal}(BP_K/K)$$ is called the Bertrandias-Payan module. We study the transfer map $$j_{L/K}: \mathcal{BP}_K\to \mathcal{BP}_L$$ (as a capitulation morphism of ideal classes) in a $$p$$-extension $$L/K$$. In the cyclic case of degree $$p$$, we prove that $$j_{L/K}$$ is injective except if $$L/K$$ is kummerian, $$p$$-ramified, non globally cyclotomic but locally cyclotomic at $$p$$ (Theorem 3.1). We give an explicit formula (Theorem 5.2) for $$|\mathcal{BP}_L^G | \cdot|\mathcal{BP}_K |^{-1}$$ and we show how its entirety depends on the torsion group $$\mathcal T_L$$ of the Galois group of the maximal abelian $$p$$-ramified pro-$$p$$-extension of $$L$$, by using a suitable $$p$$-adic logarithm.
For the entire collection see [Zbl 1365.11004].

##### MSC:
 11R37 Class field theory 11R04 Algebraic numbers; rings of algebraic integers
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