Another look at the index formulas of cyclotomic number theory

To the cyclotomic number field K generated by the roots of unity of order f we attach a Galois module which is a hybrid of the Stickelberger ideal and the group of circular units; this Galois module also admits interpretation as the universal punctured distribution of conductor f. We embed our Galois module in another naturally occurring Galois module, and prove that the index is exactly the class number of K. By avoiding even-odd splittings and the analytic class number formula, we are able to avoid the consideration of { ± 1 }-cohomology groups as well. Cohomological considerations become necessary only when making comparisons to the classical index formulas of Kummer, Hasse, Iwasawa, and Sinnott.