Construction of a certain circular unit and its applications

For an abelian number field k, let CS(k) be the group of circular units of k defined by Sinnott, and CW(k) be that suggested by Washington. In this paper, we construct an element in CW(k) for a real subfield k of conductor . We will see that the order of in the factor group CW(k)/CS(k) can be very large. As an application, we derive some information about the class number of k for special cases.