An improvement of Lenstra's criterion for Euclidean number fields: The totally real case

Niklasch, Gerhard; Queme, Roland

doi:10.4064/aa-58-2-157-168

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Cited by 3 publications

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“…We could observe that, for a given degree, maximal length of an exceptional sequence grows with unit rank. However the increase appeared to be of at most one or two units as the required bounds are multiplied by a factor between 1.5 and 2, when increasing unit rank by one (except for totally real number fields, bounds given in [13] being smaller). So it seems hopeless to show that some number fields with r 1 + r 2 > 6 are Euclidean with this method.…”

Section: Resultsmentioning

confidence: 99%

Exceptional units and Euclidean number fields

Houriet

2007

Arch. Math.

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By a result of H.W. Lenstra, one can prove that a number field is Euclidean with the aid of exceptional units. We describe two methods computing exceptional sequences, i.e., sets of units such that the difference of any two of them is still a unit. The second method is based on a graph theory algorithm for the maximum clique problem. This yielded 42 new Euclidean number fields in degrees 8, 9, 10, 11 and 12.

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Section: Resultsmentioning

confidence: 99%