Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

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 l and conductor f. Assume the GRH for sigma(K)(s). If 38(L - 1)(2)(log f)(6) log log f < f, then K is not norm-Euclidean.