Well-Rounded Twists of the Ring of Integers in Cyclic Cubic Fields

TL;DR

This paper introduces a new algorithm to identify well-rounded twists of ideals in cyclic cubic fields, extending previous work from quadratic to cubic fields, and applies it to specific families of fields.

Contribution

It develops an algorithm for detecting well-rounded twists in cyclic cubic fields and proves their existence under certain conditions, expanding the understanding of ideal structures in number fields.

Findings

Explicitly computed well-rounded twists for specific cyclic cubic fields.

Proved the existence of infinitely many fields with rings of integers admitting orthogonal well-rounded twists.

Extended methods from quadratic to cubic number fields.

Abstract

Computing well-rounded twists of ideals in number fields has been done when the field degree is $2$. In this paper, we develop a new algorithm to detect whether a basis of an ideal $\mathfrak{I}$ in a cyclic cubic field $F$ yields a well-rounded twist of $\mathfrak{I}$. We then prove that under certain conditions on a given basis of the ring of integers $\mathcal{O}_F$, the existence of its well-rounded twist is equivalent to the existence of a principal well-rounded ideal in $K$. Applying the result and the algorithm, we explicitly compute well-rounded twists of the ring of integers for cyclic cubic fields in the families of Shanks, Washington, and Kishi. In addition, we show that infinitely many fields in Shanks's family have rings of integers that admit orthogonal well-rounded twists.