Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis
Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 425-445.

En supposant que l’hypothèse de Riemann généralisée (HRG) soit vérifiée, nous montrons que les corps de nombres galoisiens de degré 3 qui sont euclidiens pour la norme sont précisément ceux dont le discriminant est l’un des entiers suivants :

Δ=72,92,132,192,312,372,432,612,672,1032,1092,1272,1572.

Une grande partie de la preuve consiste à établir le résultat plus général suivant : soit K un corps de nombres galoisien de degré premier impair et de conducteur f. Supposons que HRG soit vérifiée pour ζ K (s). Si

38(-1)2(logf)6loglogf<f,

alors K n’est pas euclidien pour la norme.

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant

Δ=72,92,132,192,312,372,432,612,672,1032,1092,1272,1572.

A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree and conductor f. Assume the GRH for ζ K (s). If

38(-1)2(logf)6loglogf<f,

then K is not norm-Euclidean.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.804
Classification : 11A05,  11R04,  11R16,  11R80,  11Y40
Mots clés : norm-Euclidean, Galois fields, cubic fields, GRH, Dirichlet characters
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     author = {Kevin J. McGown},
     title = {Norm-Euclidean {Galois} fields and the {Generalized} {Riemann} {Hypothesis}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {425--445},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {2},
     year = {2012},
     doi = {10.5802/jtnb.804},
     zbl = {1272.11005},
     mrnumber = {2950700},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.804/}
}
Kevin J. McGown. Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis. Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 425-445. doi : 10.5802/jtnb.804. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.804/

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