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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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##### References:
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