Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Kevin J. McGown[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

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 2, page 425-445
  • ISSN: 1246-7405

Abstract

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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 ℓ and conductor f . Assume the GRH for ζ K ( s ) . If 38 ( ℓ - 1 ) 2 ( log f ) 6 log log f < f , then K is not norm-Euclidean.

How to cite

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McGown, Kevin J.. "Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis." Journal de Théorie des Nombres de Bordeaux 24.2 (2012): 425-445. <http://eudml.org/doc/251115>.

@article{McGown2012,
abstract = {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 $\zeta _K(s)$. If\[ 38(\ell -1)^2(\log f)^6\log \log f&lt;f \,, \]then $K$ is not norm-Euclidean.},
affiliation = {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},
author = {McGown, Kevin J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {norm-Euclidean; Galois fields; cubic fields; GRH; Dirichlet characters; norm-Euclidean rings; number fields; cyclic extensions; cubic extensions; Dirichlet character; Riemann hypothesis},
language = {eng},
month = {6},
number = {2},
pages = {425-445},
publisher = {Société Arithmétique de Bordeaux},
title = {Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis},
url = {http://eudml.org/doc/251115},
volume = {24},
year = {2012},
}

TY - JOUR
AU - McGown, Kevin J.
TI - Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/6//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 2
SP - 425
EP - 445
AB - 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 $\zeta _K(s)$. If\[ 38(\ell -1)^2(\log f)^6\log \log f&lt;f \,, \]then $K$ is not norm-Euclidean.
LA - eng
KW - norm-Euclidean; Galois fields; cubic fields; GRH; Dirichlet characters; norm-Euclidean rings; number fields; cyclic extensions; cubic extensions; Dirichlet character; Riemann hypothesis
UR - http://eudml.org/doc/251115
ER -

References

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