Some works of Furtw\"angler and Vandiver revisited and Fermat's last theorem

TL;DR

This paper revisits classical works to develop a new cyclotomic approach to Fermat's Last Theorem and a stronger conjecture called SFLT, establishing links between solutions and field arithmetic constraints.

Contribution

It introduces a novel cyclotomic framework involving governing fields to analyze Fermat's Last Theorem and SFLT, providing new conditions and conjectures for future research.

Findings

Proves that certain prime conditions imply Fermat's Last Theorem.

Shows that solutions to SFLT impose strong constraints on field arithmetic.

Provides sufficient conditions for nonexistence of solutions, guiding future investigations.

Abstract

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.