  Let $K$ be a totally real number field. For all prime number $p\geq 5$, let
us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the
assumption that $2$ is totally ramified in $K$, we establish some results about
the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so
that the asymptotic Fermat's Last Theorem is true over $K$, criterion related
to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$
and of level the prime ideal above $2$. It is often simply testable
numerically, particularly if the narrow class number of $K$ is $1$.
Furthermore, using the modular method, we prove Fermat's Last Theorem
effectively, over some number fields whose degrees over $\mathbb{Q}$ are
$3,4,5,6$ and $8$.
