Norm-Euclidean Galois fields

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.