  The first case of Fermat's Last Theorem for a prime exponent $p$ can
sometimes be proved using the existence of local obstructions. In 1823, Sophie
Germain has obtained an important result in this direction by establishing
that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is
true for $p$. In this paper, we investigate such obstructions over number
fields. We obtain analogous results on Sophie Germain type criteria, for
imaginary quadratic fields. Furthermore, extending a well known statement over
${\bf Q}$, we give an easily testable condition which allows occasionally to
prove the first case of Fermat's Last Theorem over number fields for a prime
number $p\equiv 2 \mod 3$.
