  From some works of P. Furtw\"angler 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 Q^(1-c) =
A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex
conjugation), then Fermat's last theorem holds for p. More generally, the main
purpose of the paper is to show that the existence of nontrivial 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 future
researchs (probably of analytic or geometric nature) - This second version
includes some corrections in the English language, an in depth study of the
case p=3 (especially Theorem 8), further details on some conjectures, and some
minor mathematical improvements.
