A note on the index of irregularity (Q1073836)

Let \(\ell\) be an odd prime and let G be the Galois group over \({\mathbb{Q}}\) of the \(\ell\)-th cyclotomic field. Let \(R^-(\ell)\) be the subring of the group ring of G over \({\mathbb{Z}}/\ell {\mathbb{Z}}\) which is the -1- eigenspace for complex conjugation. There is an ideal \(I^-(\ell)\) of \(R^-(\ell)\), called the Stickelberger ideal. In the case where the group ring is taken over \({\mathbb{Z}}\), \textit{K. Iwasawa} [Ann. Math., II. Ser. 76, 171-179 (1962; Zbl 0125.020)] showed that the index \([R^- :I^-]\) of the Stickelberger ideal is \(h^-\), the relative class number of the \(\ell\)-th cyclotomic field. In the present case, the author shows that the index is \(\ell^ i\), where i is the index of irregularity of \(\ell\). He also studies various properties of the Stickelberger ideal mod \(\ell\).