## The Euclidean algorithm in quintic and septic cyclic fields

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- by Pierre Lezowski and Kevin J. McGown PDF
- Math. Comp.
**86**(2017), 2535-2549 Request permission

## 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 $f\in (2\cdot 10^{14}, 10^{50})$.

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## Additional Information

**Pierre Lezowski**- Affiliation: Université Blaise Pascal, Laboratoire de Mathématiques UMR 6620, Campus Universitaire des Cézeaux, BP 80026, 63171 Aubière Cédex, France
- MR Author ID: 988126
- Email: pierre.lezowski@math.univ-bpclermont.fr
**Kevin J. McGown**- Affiliation: California State University, Chico, Department of Mathematics and Statistics, 601 E. Main St., Chico, California 95929
- MR Author ID: 768800
- ORCID: 0000-0002-5925-801X
- Email: kmcgown@csuchico.edu
- Received by editor(s): December 1, 2015
- Received by editor(s) in revised form: March 26, 2016
- Published electronically: February 16, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 2535-2549 - MSC (2010): Primary 11A05, 11R04, 11Y40; Secondary 11R16, 11R80, 11L40, 11R32
- DOI: https://doi.org/10.1090/mcom/3169
- MathSciNet review: 3647971