The Euclidean algorithm in quintic and septic cyclic fields

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 $f\le 157$ except possibly when $f\in (2\cdot 10^{14},10^{50})$.