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On Fermat’s last theorem and the arithmetic of \({\mathbb{Z}}[\zeta _ p+\zeta _ p^{-1}]\). (English) Zbl 0654.10018
The well-known criterion of Kummer (1857) concerning the first case of Fermat’s last theorem can be formulated as follows: “If a, b, c are integers, p an odd prime, p does not divide abc, and \(a^ p+b^ p+c^ p=0,\) then \(\phi_{p-2h}(t)B_{2h}\equiv 0 (\bmod p)\quad (1\leq h\leq (p-3)/2)\) for each \(t\in G=\{a/b, a/c, b/a, b/c, c/a, c/b\}\).” Here \(B_{2h}\) means the Bernoulli numbers and \(\phi_ i(t)=\sum^{p- 1}_{j=1}(-1)^{j-1} j^{i-1} t\quad j\) is the Mirimanoff polynomial.
The main result of this paper extends the Kummer criterion into the following “complement” form: “Under assumption of the Kummer criterion and if p divides \(B_{p-1-2h}\) for some h \((1\leq h\leq (p-3)/2)\) but \(\prod^{p-1}_{k=1}(1-\zeta \quad k)^{k^{2k}}\) is not a pth power in \({\mathbb{Z}}[\zeta]\) (\(\zeta\) is a primitive pth root of unity), then \(\phi_{p-1-2h}(t)\equiv 0 (\bmod p)\) for each \(t\in G.''\)
Some further assertions on Fermat’s last theorem are mentioned. The proofs are based on a “Stickelberger like” theorem on annihilation of some elements from a special group ring \({\mathbb{Z}}[\Delta]\) on the p-Sylow subgroup of the ideal class group of \({\mathbb{Q}}(\zeta +\zeta^{-1}).\)
Reviewer: L.Skula

MSC:
11D41 Higher degree equations; Fermat’s equation
11R18 Cyclotomic extensions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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