Biquadratic fields having a non-principal Euclidean ideal class

Abstract

H. W. Lenstra [7] introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in [3] and [6]). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves [3] and C. Hsu [6].2010 Mathematics Subject Classification. 11A05.

First-Degree Prime Ideals of Biquadratic Fields Dividing Prescribed Principal Ideals

“…Biquadratic fields are numerical fields that have been studied extensively [1][2][3][4][5][6] and are currently in the spotlight as they provide examples of non-principal Euclidean ideal classes [7,8]. They are defined as numerical fields whose Galois group is the Klein group, and they can be obtained by the compositum of two quadratic fields, which is the construction we adopt in the present work.…”

“…There are three first-degree prime ideals of norm p = 7 in Z[γ]: (0, 7), (2,7), and (5,7). The first-degree prime ideals of the same norm for both Z[α] and Z[β] are (1,7) and (6,7).…”

“…There are three first-degree prime ideals of norm p = 7 in Z[γ]: (0, 7), (2,7), and (5,7). The first-degree prime ideals of the same norm for both Z[α] and Z[β] are (1,7) and (6,7). As prescribed by Theorem 3, we observe that (2, 7) and (5,7) are uniquely determined by the combinations of (1,7), (1,7) and (6, 7), (6,7) , whereas (0, 7) arises from both the combinations of (1, 7), (6,7) and (6,7), (1,7) .…”

Totally real bi-quadratic fields with large Pólya groups

“…In [4], the authors determined the Pólya groups of bi-quadratic fields that are known to have an "Euclidean ideal class". The interested reader is encouraged to look into [3] and the references listed therein to know about Euclidean ideal class.…”

Genus theory and Euclidean ideals for real biquadratic fields

“…Recently, Chattopadhyay and Muthukrishnan [2] further found two other families of biquadratic fields which have a non-principal Euclidean ideal class. Combing with Hsu's result, they got the following more systematic result.…”