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=7 2 ,9 2 ,13 2 ,19 2 ,31 2 ,37 2 ,43 2 ,61 2 ,67 2 ,103 2 ,109 2 ,127 2 ,157 2 .

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) 6 loglogf<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=7 2 ,9 2 ,13 2 ,19 2 ,31 2 ,37 2 ,43 2 ,61 2 ,67 2 ,103 2 ,109 2 ,127 2 ,157 2 .

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) 6 loglogf<f,

then K is not norm-Euclidean.

DOI : 10.5802/jtnb.804
Classification : 11A05, 11R04, 11R16, 11R80, 11Y40
Mots clés : norm-Euclidean, Galois fields, cubic fields, GRH, Dirichlet characters
McGown, Kevin J. 1

1 Department of Mathematics University of California, San Diego La Jolla, California, 92093, USA Current address : Department of Mathematics Oregon State University 368 Kidder Hall Corvallis, Oregon, 97331, USA
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McGown, Kevin J. 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. http://www.numdam.org/articles/10.5802/jtnb.804/

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