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Kummer type congruences and Stickelberger subideals. (English) Zbl 0841.11012
In connection with the first case of Fermat’s last theorem Kummer (1857) introduced a special system of congruences $$(K)\bmod \ell$$ ($$\ell$$ an odd prime) of one unknown which the Bernoulli numbers and the Mirimanoff polynomials are used in. In this paper other systems $$(K(N))$$ of congruences are defined for an integer $$N$$ $$(2\leq N\leq \ell-1)$$ possessing the property that each solution of $$(K)$$ satisfies $$(K(N))$$. Various equivalent systems to $$(K(N))$$ are introduced and a connection with special subideals $${\mathfrak B}_N$$ of the Stickelberger ideal $${\mathfrak I}$$ for the $$\ell$$th cyclotomic field is shown. The ideals $${\mathfrak B}_N$$ are constructed by means of elements used by Fueter (1922). The group indices $$[{\mathfrak I}:{\mathfrak B}_N]$$ are evaluated by constructing a $$\mathbb{Z}$$-basis of $${\mathfrak B}_N$$. The ideal $${\mathfrak B}_N$$ for $$N=2$$ is related to a modified Demyanenko matrix $$D' (\ell)$$ $$(\ell\geq 5)$$ from the paper of H. G. Folz and H. G. Zimmer [J. Symb. Comput. 4, 53-67 (1987; Zbl 0624.14001).
Reviewer: L.Skula (Brno)

##### MSC:
 11D41 Higher degree equations; Fermat’s equation 11A07 Congruences; primitive roots; residue systems 11R18 Cyclotomic extensions 11B68 Bernoulli and Euler numbers and polynomials
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