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

Abstract

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that the number field $\mathbb{Q}(\sqrt{2},\sqrt{35})$ has a non-principal Euclidean ideal.Comment: 12 page

Biquadratic fields having a non-principal Euclidean ideal class

“…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].…”

“…Later, many explicit families of number fields with cyclic class groups were unconditionally proved to possess an EIC (cf. [2], [3], [6]).…”

Genus theory and Euclidean ideals for real biquadratic fields

“…Extending the work of Graves [7], Hsu [8] explicitly constructed a family of real biquadratic fields having a non-principal Euclidean ideal class. Her result reads as follows.…”

“…Theorem 1.4 (Hsu [8]). Suppose K is a quartic field of the form Q( √ p 1 , √ q 1 q 2 ) with class number h K .…”

“…In fact, Hsu [8] also constructed a family of real cyclic quartic fields having such property. Using PARI, she computed a list of examples, and conjectured both her classes of number fields are infinite (See Remark 1.9 of [8]).…”