Relative Ideal Classes of Arbitrary Order

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.