Ramified extensions of degree p and their Hopf–Galois module structure
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 19-40.

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(π K p) pour une certaine uniformisante π K 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)={xK[G]:x𝔓 L n 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.

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(π K p) for some prime element π K 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)={xK[G]:x𝔓 L n 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.

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DOI : https://doi.org/10.5802/jtnb.1014
Classification : 11S15,  11R33,  16T05
Mots clés : Artin–Schreier equation, Galois module structure
@article{JTNB_2018__30_1_19_0,
     author = {G. Griffith Elder},
     title = {Ramified extensions of degree $p$ and their {Hopf{\textendash}Galois} module structure},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {19--40},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {1},
     year = {2018},
     doi = {10.5802/jtnb.1014},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1014/}
}
G. Griffith Elder. Ramified extensions of degree $p$ and their Hopf–Galois module structure. Journal de Théorie des Nombres de Bordeaux, Tome 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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