Examples of Norm-Euclidean Ideal Classes

Abstract

In [11], Lenstra defined the notion of Euclidean ideal class. Using a slight modification of an algorithm described in [12], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari ([5]), we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class.

Two classes of number fields with a non-principal Euclidean ideal

“…Remark 3. Note that a straightforward application of the algorithm given by Lezowski [9] determines whether the fields in Table 1 have a norm-Euclidean ideal. Table 1, we should be able to modify the hypotheses of Theorem 4 in order to obtain similar results.…”

“…Remark 3. Note that a straightforward application of the algorithm given by Lezowski [9] determines whether the fields in Table 1 have a norm-Euclidean ideal.…”

On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields

“…Let K be a number field and assume that K has a norm Euclidean ideal class C such that M C < 1. Then K satisfies the P-adic CFF property for all but finitely many P ∈ C. [25,29]). It would be nice to replace the hypothesis M C < 1 in Theorem 7.4 with the more natural hypothesis M C < 1; however, unlike the case C = O K , we do not know if an analogue of Theorem 5.4 is true for an arbitrary norm Euclidean ideal classes.…”

“…We notice that non principal Euclidean classes exist for example for fields like Q( √ −15) and Q( √ −20) (see [24, Prop. 2.1]), and Q( √ 10), Q( √ 15), Q( √ 85) (see [24, 2.5]); other examples can be found in [25].…”