  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.
