Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

McGown, Kevin J.

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 :

$$\Delta =7^2,9^2,{13}^2,{19}^2,{31}^2,{37}^2,{43}^2,{61}^2,{67}^2,{103}^2,{109}^2,{127}^2,{157}^2.$$

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 $\ell$ et de conducteur $f$. Supposons que HRG soit vérifiée pour ${\zeta }_K\left(s\right)$. Si

$$38{(\ell -1)}^2{(logf)}^{6}loglogf<f,$$

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

$$\Delta =7^2,9^2,{13}^2,{19}^2,{31}^2,{37}^2,{43}^2,{61}^2,{67}^2,{103}^2,{109}^2,{127}^2,{157}^2.$$

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 $\ell$ and conductor $f$. Assume the GRH for ${\zeta }_K\left(s\right)$. If

$$38{(\ell -1)}^2{(logf)}^{6}loglogf<f,$$

then $K$ is not norm-Euclidean.

Classification : 11A05, 11R04, 11R16, 11R80, 11Y40
Mots clés : norm-Euclidean, Galois fields, cubic fields, GRH, Dirichlet characters

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