ANNALES DE L'INSTITUT FOURIER

Groups of automorphisms of local fields of period ${p}^{M}$ and nilpotent class $  [ Groupes d’automorphismes de corps locaux de période ${p}^{M}$ et de classe de nilpotence $ ]
Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 605-635.

Soit $K$ une extension finie de ${ℚ}_{p}$ contenant une racine ${p}^{M}$-ième primitive de l’unité. Pour $1⩽s on note $K\left[s,M\right]$ la $p$-extension maximale de $K$ dont le groupe de Galois est de période ${p}^{M}$ et de classe de nilpotence $s$. En utilisant la théorie d’Artin–Schreier nilpotente et la théorie du corps des normes on donne une description explicite du groupe de Galois de $K\left[s,M\right]/K$. Comme application de ce résultat on montre que le sous-groupe de ramification du groupe de Galois absolu de $K$ de ramification supérieure $v$ agit trivialement sur $K\left[s,M\right]$ si et seulement si $v>{e}_{K}\left(M+s/\left(p-1\right)\right)-\left(1-{\delta }_{1s}\right)/p$, où ${e}_{K}$ est l’indice de ramification de $K$ et ${\delta }_{1s}$ est le symbole de Kronecker.

Suppose $K$ is a finite field extension of ${ℚ}_{p}$ containing a ${p}^{M}$-th primitive root of unity. For $1⩽s denote by $K\left[s,M\right]$ the maximal $p$-extension of $K$ with the Galois group of period ${p}^{M}$ and nilpotent class $s$. We apply the nilpotent Artin–Schreier theory together with the theory of the field-of-norms functor to give an explicit description of the Galois groups of $K\left[s,M\right]/K$. As application we prove that the ramification subgroup of the absolute Galois group of $K$ with the upper index $v$ acts trivially on $K\left[s,M\right]$ iff $v>{e}_{K}\left(M+s/\left(p-1\right)\right)-\left(1-{\delta }_{1s}\right)/p$, where ${e}_{K}$ is the ramification index of $K$ and ${\delta }_{1s}$ is the Kronecker symbol.

Reçu le : 2015-06-22
Révisé le : 2016-05-23
Accepté le : 2016-06-14
Publié le : 2017-05-31
DOI : https://doi.org/10.5802/aif.3093
Classification : 11S15,  11S20
Mots clés : Corps locaux, nombres de ramification supérieure.
@article{AIF_2017__67_2_605_0,
author = {Abrashkin, Victor},
title = {Groups of automorphisms of local fields of period $p^M$ and nilpotent class $<p$},
journal = {Annales de l'Institut Fourier},
pages = {605--635},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {2},
year = {2017},
doi = {10.5802/aif.3093},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_2017__67_2_605_0/}
}
Abrashkin, Victor. Groups of automorphisms of local fields of period $p^M$ and nilpotent class \$



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