A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$

Byott, Nigel P.

doi:10.5802/jtnb.750

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic

p

Byott, Nigel P.¹

¹ Mathematics Research Institute University of Exeter Harrison Building North Park Road Exeter EX4 4QF, UK

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 59-70.

Abstract
Résumé

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 $v_{L} (ρ) \equiv - v_{L} (𝔻_{S / R}) - 1 (mod n)$ 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.

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 $v_{L} (ρ) \equiv - v_{L} (𝔻_{S / R}) - 1 (mod n)$ 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.

MR Zbl

DOI: 10.5802/jtnb.750

Classification: 11S15
Mots-clés : Normal basis, Hopf-Galois extensions, local fields

Author's affiliations:

Byott, Nigel P. ¹

¹ Mathematics Research Institute University of Exeter Harrison Building North Park Road Exeter EX4 4QF, UK

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     title = {A valuation criterion for normal basis generators of {Hopf-Galois} extensions in characteristic $p$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {59--70},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     doi = {10.5802/jtnb.750},
     zbl = {1278.11103},
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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, Volume 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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