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Journal Articles Mathematics of Computation Year : 2014

Computation of the Euclidean minimum of algebraic number fields

Abstract

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri. With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields of degree up to 8 in any signature, especially for cyclotomic fields, and many new examples of norm-Euclidean or non-norm-Euclidean algebraic number fields. We also prove a result of independant interest concerning real quadratic fields whose Euclidean minimum is equal to 1.
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Dates and versions

hal-00632997 , version 1 (17-10-2011)
hal-00632997 , version 2 (02-10-2012)

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Pierre Lezowski. Computation of the Euclidean minimum of algebraic number fields. Mathematics of Computation, 2014, 83, pp.1397-1426. ⟨10.1090/S0025-5718-2013-02746-9⟩. ⟨hal-00632997v2⟩
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