The Euclidean algorithm in quintic and septic cyclic fields
Abstract
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=
11
4
,
31
4
,
41
4
; (2) a cyclic number field of degree
7
is norm-Euclidean if and only if
Δ=
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∈(2⋅
10
14
,
10
50
)
.