The determination of all imaginary, quartic, abelian number fields with class number $1$

In this paper, it is proved that there are just seven imaginary number fields, quartic cyclic over the rational field, and having class number 1. These are the quartic, cyclic imaginary subfields of the cyclotomic fields generated by the fth roots of unity, where f is 16 or is a prime less than 100. This completes the list of imaginary, quartic, abelian number fields with class number 1. There are 54 such fields, with maximal conductor 67.163.