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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].

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