The Euclidean algorithm in quintic and septic cyclic fields

TL;DR

Under the assumption of GRH, the paper classifies norm-Euclidean cyclic fields of degrees 5 and 7, identifies specific discriminants, and shows the non-existence of such fields for higher degrees, with some unconditional results in cubic cases.

Contribution

The paper provides a conditional classification of norm-Euclidean cyclic fields of degrees 5 and 7, and establishes non-existence results for higher degrees, including computational methods and unconditional bounds for cubic fields.

Findings

Cyclic degree 5 fields are norm-Euclidean iff discriminant is 11^4, 31^4, or 41^4.

Cyclic degree 7 fields are norm-Euclidean iff discriminant is 29^6 or 43^6.

No norm-Euclidean cyclic fields of degrees 19, 31, 37, 43, 47, 59, 67, 71, 73, 79, 97.

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 $\Delta=11^4,31^4,41^4$; (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 $f\leq 157$ except possibly when $f\in(2\cdot 10^{14}, 10^{50})$.