no. 2
Groups of automorphisms of local fields of period p M and nilpotent class <p
[Groupes d’automorphismes de corps locaux de période p M et de classe de nilpotence <p]
Abrashkin, Victor1, 2
1 Department of Mathematical Sciences Durham University Lower Mountjoy, Stockton Rd DH1 3LE (UK)
2 Steklov Institute Gubkina str. 8 119991 Moscow (Russia)
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<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.

Suppose K is a finite field extension of ℚ p containing a p M -th primitive root of unity. For 1⩽s<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.

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DOI : 10.5802/aif.3093
Classification : 11S15, 11S20
Keywords: local fields, upper ramification numbers
Mot clés : Corps locaux, nombres de ramification supérieure.

Abrashkin, Victor 1, 2

1 Department of Mathematical Sciences Durham University Lower Mountjoy, Stockton Rd DH1 3LE (UK)
2 Steklov Institute Gubkina str. 8 119991 Moscow (Russia)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Groups of automorphisms of local fields of period $p^M$ and nilpotent class $<p$},
     journal = {Annales de l'Institut Fourier},
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Abrashkin, Victor. Groups of automorphisms of local fields of period $p^M$ and nilpotent class $

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