Genus theory and Euclidean ideals for real biquadratic fields

TL;DR

This paper explores Euclidean ideals in real biquadratic fields using genus theory, expanding known families and handling cases with larger class numbers, revealing new properties of their Hilbert class fields.

Contribution

It introduces a new approach using genus theory to identify larger families of real biquadratic fields with Euclidean ideals, applicable to broader class number cases.

Findings

Identifies larger families of real biquadratic fields with Euclidean ideals.

Shows that for these families, the Hilbert class field is non-abelian over Q when class number  4.

Extends methods to cases where class number is a power of two,  not just two.

Abstract

In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean ideals; and the conditions for our family seem to be more efficient for the computations. Moreover, the previous approaches mainly focus on the case if $h_K=2$, while the present approach can also deal with the general case when $h_K=2^t (t\geq1)$, where $h_K$ denotes the ideal class number of $K$. In particular, if $h_K\geq 4$, it shows that $H(K)$, the Hilbert class field of $K$, is always non-abelian over $\mathbb{Q}$ for the family of $K$ given in this paper having Euclidean ideals, whereas the previous approaches always requires that $H(K)$ is abelian over $\mathbb{Q}$ explicitly or implicitly. Finally, some open questions have also been listed for further research.