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Kummer type system of congruences and bases of Stickelberger subideals. (English) Zbl 0903.11007
Trying to solve the first case of Fermat’s Last Theorem for an odd prime \(l,\) Kummer (1857) introduced the following system of congruences: \[ \begin{aligned} \phi _{l-1}(t)&\equiv 0 \pmod l,\\ B_{2m}\phi _{l-2m}&\equiv 0\pmod l\quad (1\leq m\leq (l-3)/2), \end{aligned} \tag{K} \] where \(B_{2m}\) denotes the Bernoulli number and \(\phi _k (x) = \sum _{v=1}^{l-1}v^{k-1} x^v\) is the Mirimanoff polynomial.
For an integer \(N\), \(2\leq N\leq l-1,\) T. Agoh and L. Skula [Acta Arith. 75, 235-250 (1996; Zbl 0841.11012)] considered the system \(K(N)\), which is created substituting the numbers \(B^{(N)}_{2m}=B_{2m}(1-N^{2m})/2m\) for the Bernoulli numbers \(B_{2m}\) in (K). If in (K) the Bernoulli numbers \(B_{2m}\) are replaced by the numbers \(B_{2m}^{(M,N)} =B_{2m}(1-M^{2m})(1-N^{2m})/2m\) for fixed integers \(M\), \(N\;(2\leq M,N\leq l-1),\) we get the system \( (K(M,N))\) investigated in this paper.
Let \(R\), \(R_l\) be the group ring of the Galois group of the \(l\)th cyclotomic field over the ring \(\mathbb{Z}\) of integers or over the ring \(\mathbb{Z}/l\mathbb{Z}\) of residue classes modulo \(l,\) respectively. L. Skula [Comment. Math. Univ. St. Pauli 31, 89-97 (1982; Zbl 0496.10006)] defined a special map from \(R\;(R_l)\) into the set of polynomials \(\mathbb{Z}[t]\) and showed that the system (K) corresponds to the Stickelberger ideal \(I\) (the Stickelberger ideal \(I(l)\) of \(R_l\)). The authors describe the ideal \(I_{M,N}(l)\) of \(R_l\) and \(B_{M,N}\) of \(R\) corresponding to the system \(K(M,N),\) and, in this way they extend the result concerning the system \( (K(N))\) with corresponding ideals \(I_N(l)\) and \(B_N\) from the mentioned paper of Agoh and Skula. Some bases of ideals \(I_{M,N}(l)\) and \(B_{M,N}\) are constructed in this article.
For an integer \(X\), \(2\leq X\leq l-1,\) put \[ \omega (X)=\begin{cases} (X^{f/2}+1)^{(l-1)/2}&\text{if }f\text{ is even,}\\ (X^{f}-1)^{(l-1)/2f}&\text{if }f\text{ is odd,}\\ \end{cases} \] where \(f\) denotes the order of \(X\) mod \(l.\) It is shown in this paper (Theorem 5.8) that \[ [R': B_{M,N}] =\frac {\omega (M)\omega (N)}{l}h^-, \] where \(h^-\) is the first factor of the class number of the \(l\)th cyclotomic field and \(R'\) is a special subring of \(R\).
Reviewer: L.Skula (Brno)

11B68 Bernoulli and Euler numbers and polynomials
16S34 Group rings
11R54 Other algebras and orders, and their zeta and \(L\)-functions
11R23 Iwasawa theory
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