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

Remarques sur le premier cas du th\'eor\`eme de Fermat sur les corps de nombres