THE EUCLIDEAN ALGORITHM IN QUINTIC AND SEPTIC CYCLIC FIELDS

Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree 5 is norm Euclidean if and only if Delta = 11(4), 31(4), 41(4); (2) a cyclic number field of degree 7 is norm-Euclidean if and only if Delta = 29(6), 43(6); (3) there are no norm-Euclidean cyclic number fields of degrees 19, 31, 37, 43, 47, 59, 67, 71, 73, 79, 97. Our proofs contain a large computational component, including the calculation of the Euclidean minimum in some cases; the correctness of these calculations does not depend upon the GRH. Finally, we improve on what is known unconditionally in the cubic case by showing that any norm-Euclidean cyclic cubic field must have conductor f <= 157 except possibly when f E (2 center dot 10(14), 10(50)).