# The Euclidean algorithm in quintic and septic cyclic fields

2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
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 $f\in(2\cdot 10^{14}, 10^{50})$.
Type de document :
Article dans une revue
Mathematics of Computation, American Mathematical Society, 2017, 86 (307), pp.2535--2549. 〈10.1090/mcom/3169〉

Littérature citée [19 références]

https://hal.archives-ouvertes.fr/hal-01258906
Contributeur : Pierre Lezowski <>
Soumis le : jeudi 28 avril 2016 - 15:06:06
Dernière modification le : dimanche 11 mars 2018 - 20:28:01
Document(s) archivé(s) le : mardi 15 novembre 2016 - 16:39:28

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quintic_septic_160319.pdf
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Pierre Lezowski, Kevin Mcgown. The Euclidean algorithm in quintic and septic cyclic fields. Mathematics of Computation, American Mathematical Society, 2017, 86 (307), pp.2535--2549. 〈10.1090/mcom/3169〉. 〈hal-01258906v2〉

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