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The Euclidean algorithm in quintic and septic cyclic fields

hal.structure.identifierLaboratoire de Mathématiques Blaise Pascal [LMBP]
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
dc.contributor.authorLEZOWSKI, Pierre
hal.structure.identifierDepartment of Mathematics and Statistics [Chico]
dc.contributor.authorMCGOWN, Kevin
dc.date.accessioned2024-04-04T03:14:52Z
dc.date.available2024-04-04T03:14:52Z
dc.date.created2016-03-19
dc.date.issued2017-09-01
dc.identifier.issn0025-5718
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/194103
dc.description.abstractEnConditionally 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 $f\in(2\cdot 10^{14}, 10^{50})$.
dc.language.isoen
dc.publisherAmerican Mathematical Society
dc.title.enThe Euclidean algorithm in quintic and septic cyclic fields
dc.typeArticle de revue
dc.identifier.doi10.1090/mcom/3169
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.identifier.arxiv1601.03433
bordeaux.journalMathematics of Computation
bordeaux.page2535--2549
bordeaux.volume86
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue307
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01258906
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01258906v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Mathematics%20of%20Computation&rft.date=2017-09-01&rft.volume=86&rft.issue=307&rft.spage=2535--2549&rft.epage=2535--2549&rft.eissn=0025-5718&rft.issn=0025-5718&rft.au=LEZOWSKI,%20Pierre&MCGOWN,%20Kevin&rft.genre=article


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