On the Stickelberger Ideal and the Circular Units of a Cyclotomic Field

By a cyclotomic field, we shall mean a subfield of the complex numbers C generated over the rational numbers Q by a root of unity. Let k be an imaginary cyclotomic field. Let Cn = e2ri/ for any integer n > 1. There is then a unique integer m > 2, m t 2 mod 4, such that k Q(Qm); we call m the conductor of k. We consider in this paper two objects associated with k: the Stickelberger ideal S and the circular units C. Let G be the Galois group of k over Q, and let R = Z[G] be a group ring of G over the ordinary integers Z. S is then an ideal of R; C is a subgroup of the group of units Eof k. Let j denote the element of G induced by complex conjugation. If A is any G-module, we denote by A+ the set of elements a in A for which ja = a, and by Athe set of elements a in A for which ja= -a. Our main result will be a computation of the indices [R-: S-] and [E+: C+]. We now state our result precisely. Let h denote the class number of k, h+ the class number of k+ (the maximal totally real subfield of k), and put hh/h+. Then we prove the following.