\'Equation de Fermat et nombres premiers inertes

TL;DR

This paper proves the second case of Fermat's Last Theorem over certain number fields using classical and Faltings' methods, focusing on primes that are regular and inert in the field.

Contribution

It establishes the second case of Fermat's Last Theorem over number fields for primes that are $K$-regular and inert, extending classical results to new algebraic settings.

Findings

Proves the second case of Fermat's Last Theorem over specific number fields.

Utilizes Faltings' theorem to control rational points on curves.

Provides practical criteria for Fermat's Last Theorem over imaginary quadratic fields.

Abstract

Let $K$ be a number field and $p$ a prime number $\geq 5$. Let us denote by $\mu_p$ the group of the $p$th roots of unity. We define $p$ to be $K$-regular if $p$ does not divide the class number of the field $K(\mu_p)$. Under the assumption that $p$ is $K$-regular and inert in $K$, we establish the second case of Fermat's Last Theorem over $K$ for the exponent $p$. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over $K$ has a finite number of $K$-rational points. Moreover, if $K$ is an imaginary quadratic field, other than ${\bf Q}(\sqrt{-3})$, we deduce a statement which allows often in practice to prove Fermat's Last Theorem over $K$ for the $K$-regular exponents.