  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 ($y'' = 6y^2$). 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.
