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The Euclidean algorithm in quintic and septic cyclic fields


Authors: Pierre Lezowski and Kevin J. McGown
Journal: Math. Comp. 86 (2017), 2535-2549
MSC (2010): Primary 11A05, 11R04, 11Y40; Secondary 11R16, 11R80, 11L40, 11R32
DOI: https://doi.org/10.1090/mcom/3169
Published electronically: February 16, 2017
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Abstract: Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree $ 5$ is norm-Euclidean if and only if $ \Delta =11^4,31^4,41^4$; (2) a cyclic number field of degree $ 7$ is norm-Euclidean if and only if $ \Delta =29^6,43^6$; (3) there are no norm-Euclidean cyclic number fields of degrees $ 19$, $ 31$, $ 37$, $ 43$, $ 47$, $ 59$, $ 67$, $ 71$, $ 73$, $ 79$, $ 97$.

Our proofs contain a large computational component, including the calculation of the Euclidean minimum in some cases; the correctness of these calculations does not depend upon the GRH. Finally, we improve on what is known unconditionally in the cubic case by showing that any norm-Euclidean cyclic cubic field must have conductor $ f\leq 157$ except possibly when $ f\in (2\cdot 10^{14}, 10^{50})$.


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Additional Information

Pierre Lezowski
Affiliation: Université Blaise Pascal, Laboratoire de Mathématiques UMR 6620, Campus Universitaire des Cézeaux, BP 80026, 63171 Aubière Cédex, France
Email: pierre.lezowski@math.univ-bpclermont.fr

Kevin J. McGown
Affiliation: California State University, Chico, Department of Mathematics and Statistics, 601 E. Main St., Chico, California 95929
Email: kmcgown@csuchico.edu

DOI: https://doi.org/10.1090/mcom/3169
Received by editor(s): December 1, 2015
Received by editor(s) in revised form: March 26, 2016
Published electronically: February 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society