LEZOWSKI, Pierre; MCGOWN, Kevin

doi:10.1090/mcom/3169

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LEZOWSKI, Pierre
Laboratoire de Mathématiques Blaise Pascal [LMBP]
Lithe and fast algorithmic number theory [LFANT]

MCGOWN, Kevin
Department of Mathematics and Statistics [Chico]

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Mathematics of Computation. 2017-09-01, vol. 86, n° 307, p. 2535--2549

American Mathematical Society

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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 $f\in(2\cdot 10^{14}, 10^{50})$.< Réduire

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The Euclidean algorithm in quintic and septic cyclic fields

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