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On a special ideal contained in the Stickelberger ideal. (English) Zbl 0861.11063
Let \(l\) be an odd prime and let \(G\) be a cyclic group of order \(l-1\) with generator \(s\). Let \(r\) be a primitive root mod \(l\) and let \(r_i\) denote the smallest positive residue mod \(l\) of \(r^i\). Let \[ R^*= \left\{\alpha \in\mathbb{Z} [G] \mid (1+s^{(l-1)/2}) \alpha \in\mathbb{Z} \cdot \sum^{l-2}_{i=0} s^i \right\}. \] Let \(I\) be the Stickelberger ideal of \(R\). In fact, \(I \subseteq R^*\) and W. Sinnott [Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)] showed that \([R^*:I] = h^-\), the relative class number of the \(l\)-th cyclotomic field. Let \(\kappa_0 = \sum s^i\), where the sum is over those \(i\) such that \(r_{-i} > l/2\), and let \({\mathcal B} = \kappa_0 \mathbb{Z} [G]\). The author shows that \([R^*: {\mathcal B}]\) equals the determinant of a matrix introduced by F. Hazama [J. Number Theory 34, 174-177 (1990; Zbl 0697.12003)], who showed that this determinant equals \(h^-\) times an elementary factor. The author studies \(R^*\) and \({\mathcal B} \bmod l\) and relates the corresponding index to the index of irregularity and Wieferich primes. He also gives relations with a system of congruences introduced by G. Benneton [Ann. Sci. Univ. Besançon, III. Ser., Math. 7 (1974; Zbl 0348.10010)].

11R18 Cyclotomic extensions
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