  We prove the rationality of a $\k$-form $X$ of the product $S$ of projective
spaces provided the existence of a $\k$-point on $X$. The method of the proof
is to find a Galois-invariant birational projection of $S$ to the projective
space. This method also allows to prove the quasitriviality of the forms of the
hyperplane sections of some Segre varieties.
