THE EUCLIDEAN ALGORITHM IN QUINTIC AND SEPTIC CYCLIC FIELDS

被引:3
作者
Lezowski, Pierre [1 ]
Mcgown, Kevin J. [2 ]
机构
[1] Univ Blaise Pascal, Lab Math, UMR 6620, Campus Univ Cezeaux,BP 80026, F-63171 Aubiere, France
[2] Calif State Univ Chico, Dept Math & Stat, 601 E Main St, Chico, CA 95929 USA
关键词
CUBIC FIELDS; NUMBER-FIELDS; NON-RESIDUE;
D O I
10.1090/mcom/3169
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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)).
引用
收藏
页码:2535 / 2549
页数:15
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