Groups of automorphisms of local fields of period p M and nilpotent class <p
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 605-635
Suppose K is a finite field extension of p containing a p M -th primitive root of unity. For 1s<p denote by K[s,M] 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[s,M]/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[s,M] iff v>e K (M+s/(p-1))-(1-δ 1s )/p, where e K is the ramification index of K and δ 1s is the Kronecker symbol.
Soit K une extension finie de p contenant une racine p M -ième primitive de l’unité. Pour 1s<p on note K[s,M] 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[s,M]/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[s,M] si et seulement si v>e K (M+s/(p-1))-(1-δ 1s )/p, où e K est l’indice de ramification de K et δ 1s est le symbole de Kronecker.
Received : 2015-06-22
Revised : 2016-05-23
Accepted : 2016-06-14
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3093
Classification:  11S15,  11S20
Keywords: local fields, upper ramification numbers
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {605-635},
     doi = {10.5802/aif.3093},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_2_605_0}
}
Groups of automorphisms of local fields of period $p^M$ and nilpotent class $
            
          

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