An observation on the first case of Fermat's last theorem (Q1120614)

The main result of this paper is contained in the following Theorem: ``Let p be an odd prime and let x,y,z be integers with the properties \[ x^ p+y^ p+z^ p=0,\quad xyz\not\equiv 0 (mod p) \] (thus, the first case of Fermat's Last Theorem fails). Let M be an imaginary proper subfield of the cyclotomic field \(L={\mathbb{Q}}(\xi_ p)\). Put for \(T\in L:\quad \Phi (T)=N_{L/M}(T+\xi_ p)-N_{L/M}(T+\xi_ p^{-1}).\) If p does not divide the relative class number \(h^-_ M\) of the field M, then \[ \Phi (T)\equiv 0 (mod p) \] for each \(T\in \{t,1/t,-1/(1+t),- (1+t),-t/(1+t),-(1+1/t)\}\) where \(t=x/y.''\) This theorem is used for the derivation of the Corollary: ``If the first case of Fermat's Last Theorem for a prime \(p\equiv 1 (mod 3)\) fails, then there exists \(j=1,3,...,(p-4)/3\) such that p divides the Bernoulli number \(B_{3j+1}.''\) This paper is completed by a table of all solutions of the congruence \(\Phi\) (T)\(\equiv 0 (mod p)\) when \(p\equiv -1 (mod 4)\) and \(M={\mathbb{Q}}(\sqrt{-p})\) (7\(\leq p\leq 199)\).