A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p
Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 59-70.

Soit S/R une extension finie d’anneaux de valuation discrète de caractéristique p>0, et supposons que l’extension correspondante L/K des corps de fractions soit séparable et H-Galoisienne pour une K-algèbre de Hopf H. Soit 𝔻 S/R la différente de S/R. Nous montrons que si S/R est totalement ramifiée et que son degré n est une puissance de p alors tout élément ρ de L avec v L (ρ)-v L (𝔻 S/R )-1(modn) engendre L comme H-module. Ce critère est le meilleur possible. Ces résultats généralisent à la situation Hopf-Galoisienne un travail récent de G. G. Elder pour les extensions Galoisiennes.

Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let 𝔻 S/R be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element ρ of L with v L (ρ)-v L (𝔻 S/R )-1(modn) generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

DOI : 10.5802/jtnb.750
Classification : 11S15
Mots clés : Normal basis, Hopf-Galois extensions, local fields
Byott, Nigel P. 1

1 Mathematics Research Institute University of Exeter Harrison Building North Park Road Exeter EX4 4QF, UK
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Byott, Nigel P. A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$. Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 59-70. doi : 10.5802/jtnb.750. http://www.numdam.org/articles/10.5802/jtnb.750/

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