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Conditionally on the Generalized Riemann Hypothesis (GRH), we prove thefollowing results: (1) a cyclic number field of degree $5$ is norm-Euclidean ifand only if $\Delta=11^4,31^4,41^4$; (2) a cyclic number field of degree $7$ isnorm-Euclidean if and only if $\Delta=29^6,43^6$; (3) there are nonorm-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 calculationof the Euclidean minimum in some cases; the correctness of these calculationsdoes not depend upon the GRH. Finally, we improve on what is knownunconditionally in the cubic case by showing that any norm-Euclidean cycliccubic field must have conductor $f\leq 157$ except possibly when $f\in(2\cdot10^{14}, 10^{50})$.

The Euclidean algorithm in quintic and septic cyclic fields