Ramified extensions of degree $p$ and their Hopf–Galois module structure

G. Griffith Elder

doi:10.5802/jtnb.1014

Ramified extensions of degree

p

and their Hopf–Galois module structure

G. Griffith Elder ¹

¹ Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182-0243, USA

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 19-40.

Abstract
Résumé

Cyclic, ramified extensions $L / K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. They are defined by an Artin–Schreier equation, unless $char (K) = 0$ and $L = K (\sqrt[p]{π_{K}})$ for some prime element $π_{K} \in K$ . Moreover, through the work of Bertrandias–Ferton ( $char (K) = 0$ ) and Aiba ( $char (K) = p$ ), much is known about the Galois module structure of the ideals in such extensions: the structure of each ideal $𝔓_{L}^{n}$ as a module over its associated order $𝔄_{K [G]} (n) = {x \in K [G] : x 𝔓_{L}^{n} \subseteq 𝔓_{L}^{n}}$ where $G = Gal (L / K)$ . The purpose of this paper is to extend these results to separable, ramified extensions of degree $p$ that are not Galois.

Les extensions cycliques ramifiées $L / K$ de degré $p$ d’un corps local dont la caractéristique résiduelle est $p$ sont plutôt bien comprises. Elles sont définies par une équation d’Artin–Schreier sauf lorsque $char (K) = 0$ et $L = K (\sqrt[p]{π_{K}})$ pour une certaine uniformisante $π_{K} \in K$ . De plus, depuis les travaux de Bertrandias–Ferton ( $char (K) = 0$ ) puis Aiba ( $char (K) = p$ ), plusieurs résultats sont connus sur la structure galoisienne des idéaux de telles extensions : on sait par exemple décrire la structure de chaque idéal $𝔓_{L}^{n}$ comme module sur son ordre associé $𝔄_{K [G]} (n) = {x \in K [G] : x 𝔓_{L}^{n} \subseteq 𝔓_{L}^{n}}$ , où $G = Gal (L / K)$ . Le but de cet article est d’étendre ces résultats aux extensions séparables et ramifiées de degré $p$ qui ne sont pas galoisiennes.

Received: 2015-10-18
Revised: 2017-04-18
Accepted: 2017-04-18
Published online: 2018-05-14

MR Zbl

DOI: 10.5802/jtnb.1014

Classification: 11S15, 11R33, 16T05
Mots-clés : Artin–Schreier equation, Galois module structure

Author's affiliations:

G. Griffith Elder ¹

¹ Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182-0243, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Ramified extensions of degree $p$ and their {Hopf{\textendash}Galois} module structure},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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G. Griffith Elder. Ramified extensions of degree $p$ and their Hopf–Galois module structure. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 19-40. doi : 10.5802/jtnb.1014. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1014/

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