Some works of Furtwängler and Vandiver revisited and the Fermat last theorem
Abstract
From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi / (q-1))) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi / (q-1))) , we have the relation Q^(1-c) = A^p.(alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then the Fermat theorem holds for p. More generally, the main purpose of the paper is to show that the existence of solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(exp(2 i pi / (q-1))). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for further researchs (probably of analytic or geometric nature).
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