EXAMPLES OF NORM-EUCLIDEAN IDEAL CLASSES

Abstract

In this paper, we study the notion of norm-Euclidean ideal class, which was defined by Lenstra [Euclidean ideal classes, Astérisque 61 (1979) 121–131]. Using a slight modification of an algorithm determining among other properties the Euclidean minimum described in [P. Lezowski, Computation of the Euclidean minimum of algebraic number fields, preprint (2011); http://hal.archives-ouvertes.fr/hal-00632997/en/], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari [The Euclidean condition in pure cubic and complex quartic fields, Math. Comput. 33 (1979) 389–398], we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class. Finally, we show that [Formula: see text], which is known to admit a non-principal Euclidean ideal class, has no norm-Euclidean ideal class.