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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)

##### MSC:
 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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