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