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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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