  Assuming the Generalized Riemann Hypothesis (GRH), we show that the
norm-Euclidean Galois cubic fields are exactly those with discriminant
$\Delta=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A
large part of the proof is in establishing the following more general result:
Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$.
Assume the GRH for $\zeta_K(s)$. If $38(\ell-1)^2(\log f)^6\log\log f<f$, then
$K$ is not norm-Euclidean.
