Analysis of the classical cyclotomic approach to fermat's last theorem

TL;DR

This paper revisits classical cyclotomic methods for Fermat's Last Theorem, emphasizing purely class field theory proofs and critiquing the reliance on computational approaches, while proposing alternative local and diophantine methods.

Contribution

It demonstrates that classical results can be proved using only class field theory, highlighting potential inefficiencies in historical computational methods and suggesting new directions for diophantine studies.

Findings

Class field theory suffices for classical cyclotomic proofs

Computational methods may be too local for a complete proof

Proposes alternative diophantine approaches using radicals

Abstract

We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However we use the derivation method of Eichler as a prerequisite for our purpose, a method which is also local but more effective. Then we propose some modest ways of study in a more diophantine context using radicals; this point of view would require further nonalgebraic investigations.