Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant
A large part of the proof is in establishing the following more general result: Let be a Galois number field of odd prime degree and conductor . Assume the GRH for . If
then is not norm-Euclidean.
En supposant que l’hypothèse de Riemann généralisée (HRG) soit vérifiée, nous montrons que les corps de nombres galoisiens de degré qui sont euclidiens pour la norme sont précisément ceux dont le discriminant est l’un des entiers suivants :
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 . Supposons que HRG soit vérifiée pour . Si
alors n’est pas euclidien pour la norme.
Keywords: norm-Euclidean, Galois fields, cubic fields, GRH, Dirichlet characters
@article{JTNB_2012__24_2_425_0, author = {McGown, Kevin J.}, 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 = {http://archive.numdam.org/articles/10.5802/jtnb.804/} }
TY - JOUR AU - McGown, Kevin J. TI - Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 425 EP - 445 VL - 24 IS - 2 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.804/ DO - 10.5802/jtnb.804 LA - en ID - JTNB_2012__24_2_425_0 ER -
%0 Journal Article %A McGown, Kevin J. %T Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis %J Journal de théorie des nombres de Bordeaux %D 2012 %P 425-445 %V 24 %N 2 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.804/ %R 10.5802/jtnb.804 %G en %F JTNB_2012__24_2_425_0
McGown, Kevin J. Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 425-445. doi : 10.5802/jtnb.804. http://archive.numdam.org/articles/10.5802/jtnb.804/
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