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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)].

MSC:
 11R18 Cyclotomic extensions
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