Le groupo\"{i}de de Galois de $P\mathrm{\_}1$ et son irr\'{e}ductibilit\'{e}

In this article, the Galois groupoid of the first Painlev'{e} equation is computed. This computation use E. Cartan’s classification of structural equations of pseudogroups acting on $C^2$ and the degeneration of the first Painlev'{e} equation on an elliptic equation ($y^{\mathrm{\prime }\mathrm{\prime }}=6y^2$). A definition of reducibility for singular holomorphic foliations is proposed. A characterisation of reducible foliations on their Galois groupoid is given and applied to prove the foliation irreducibility of the first Painlev'{e} equation.