  Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and
$\mathbb{Q}^{tr}$ be the subfield of $\overline{\mathbb{Q}}$ obtained by taking
the union of all totally real number fields. For any prime $p\geq 3$, let
$F_p/\mathbb{Q}$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop
has shown that the field $\mathbb{Q}^{tr}$ is large. In particular, the set
$F_p(\mathbb{Q}^{tr})$ of the points of $F_p$ rational over $\mathbb{Q}^{tr}$
is infinite. How to explicit non-trivial points $(xyz\neq 0$) in
$F_p(\mathbb{Q}^{tr})$ ? If one has $p\geq 5$, it seems that the only points
already known in $F_p(\mathbb{Q}^{tr})$ are those of $F_p(\mathbb{Q})$ and they
are trivial. In this paper, we investigate this question in case $p=5$. There
are no totally real fields whose degree over $\mathbb{Q}$ is at most $5$ over
which $F_5$ has non-trivial points. We propose here to explicit infinitely many
points of $F_5$ rational over totally real fields of degree $6$ over
$\mathbb{Q}$.
