  Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that
for a given $\ell$ there are only finitely many such fields that are
norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have
been identified, but for $\ell\neq 2$, little else is known. We give the first
upper bounds on the discriminants of such fields when $\ell>2$. Our methods
lead to a simple algorithm which allows one to generate a list of candidate
norm-Euclidean fields up to a given discriminant, and we provide some
computational results.
