Suppose is a finite field extension of containing a -th primitive root of unity. For denote by the maximal -extension of with the Galois group of period and nilpotent class . 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 . As application we prove that the ramification subgroup of the absolute Galois group of with the upper index acts trivially on iff , where is the ramification index of and is the Kronecker symbol.
Soit une extension finie de contenant une racine -ième primitive de l’unité. Pour on note la -extension maximale de dont le groupe de Galois est de période et de classe de nilpotence . 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 . Comme application de ce résultat on montre que le sous-groupe de ramification du groupe de Galois absolu de de ramification supérieure agit trivialement sur si et seulement si , où est l’indice de ramification de et est le symbole de Kronecker.
Revised:
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Keywords: local fields, upper ramification numbers
Mot clés : Corps locaux, nombres de ramification supérieure.
Abrashkin, Victor 1, 2
@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{\textquoteright}institut Fourier}, volume = {67}, number = {2}, year = {2017}, doi = {10.5802/aif.3093}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3093/} }
TY - JOUR AU - Abrashkin, Victor TI - Groups of automorphisms of local fields of period $p^M$ and nilpotent class $ JO - Annales de l'Institut Fourier PY - 2017 SP - 605 EP - 635 VL - 67 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3093/ DO - 10.5802/aif.3093 LA - en ID - AIF_2017__67_2_605_0 ER -
%0 Journal Article %A Abrashkin, Victor %T Groups of automorphisms of local fields of period $p^M$ and nilpotent class $ %J Annales de l'Institut Fourier %D 2017 %P 605-635 %V 67 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3093/ %R 10.5802/aif.3093 %G en %F 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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