Relative Ideal Classes of Arbitrary Order

TL;DR

This paper extends a technique for finding ideal classes of arbitrary order to various families of number fields, demonstrating the existence of infinitely many fields with such classes, including non-Galois cases.

Contribution

It adapts a known method to non-Galois fields and multiple families, proving the infinitude of fields with ideal classes of arbitrary order.

Findings

Infinitely many cyclic sextic fields contain a relative ideal class of order r.

Infinitely many cyclic quartic fields contain a relative ideal class of order r.

A family of non-Galois cubic fields contains infinitely many with an ideal class of order coprime to three.

Abstract

We adapt a known technique for searching for ideal classes of arbitrary order and then apply it to three families of number fields. We show that a family of cyclic sextic number fields has infinitely many fields in it that contain a relative ideal class of order $r,$ where $r$ is a positive integer relatively prime to the degree of the extension. We then show that the same holds true for a family of cyclic quartic number fields. Though the technique is traditionally applied to Galois extensions, we show how it may be adapted to handle a family of non-Galois cubic number fields and prove that this family contains infinitely many fields with an ideal class of arbitrary order relatively prime to three.