Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Abstract

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.