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Article Dans Une Revue Mathematics of Computation Année : 2017

The Euclidean algorithm in quintic and septic cyclic fields

Résumé

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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Dates et versions

hal-01258906 , version 1 (19-01-2016)
hal-01258906 , version 2 (28-04-2016)

Identifiants

Citer

Pierre Lezowski, Kevin J. Mcgown. The Euclidean algorithm in quintic and septic cyclic fields. Mathematics of Computation, 2017, 86 (307), pp.2535--2549. ⟨10.1090/mcom/3169⟩. ⟨hal-01258906v2⟩
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