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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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