2016/01/13 by Lezowski, Pierre, McGown, Kevin J. · 1 citation
#11A05 #11R04 #11Y40 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1601.03433
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 Δ=114,314,414; (2) a cyclic number field of degree 7 is norm-Euclidean if and only if Δ=296,436; (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∈(2⋅ 1014, 1050).