On Fermat's last theorem and the arithmetic of \({\mathbb{Z}}[\zeta _ p+\zeta _ p^{-1}]\)
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 \({\mathbb{Q}}(\zeta +\zeta^{-1}).\)
scientific article; zbMATH DE number 4067005