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Some bases of the Stickelberger ideal. (English) Zbl 0798.11044
Let $$I$$ denote the Stickelberger ideal of the group ring $$R = \mathbb{Z} [\text{Gal} (K/ \mathbb{Q})]$$, where $$K$$ is the $$p^{n+1}$$-th cyclotomic field $$(p$$ an odd prime, $$n \geq 0)$$. The author studies various bases of the $$\mathbb{Z}$$-module $$I$$. As an application he provides a new proof for Sinnott’s formula $$[R^*:I] = h^ -$$, where $$R^*$$ is a certain subring of $$R$$ and $$h^ -$$ denotes the relative class number of $$K$$ [see W. Sinnott, Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]. Moreover he gives both a $$\pi$$-adic version $$(\pi$$ any rational prime) and a modulo $$p$$ version of this index formula. The latter reads $$[R^* (p) : I(p)] = p^{i(p)}$$, where $$R^* (p)$$ and $$I(p)$$ are reductions mod $$p$$ of $$R^*$$ and $$I$$, respectively, and $$i(p)$$ denotes the index of irregularity of $$p$$.
As a further application the author proves the equivalence of several systems of congruences mod $$p$$ introduced by various authors in the study of the Fermat equation $$x^ p + y^ p = z^ p$$. One of these systems is the well-known Kummer system involving Bernoulli numbers and Mirimanoff polynomials. In this piece of work the author continues his earlier study [see Comment. Math. Univ. St. Pauli 31, 89-97 (1982; Zbl 0496.10006)].

##### MSC:
 11R18 Cyclotomic extensions 11D41 Higher degree equations; Fermat’s equation 11R29 Class numbers, class groups, discriminants 16S34 Group rings
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