Correspondances compatibles avec une relation binaire, relevement d'extensions de groupe de Galois L3(2) et probleme de Noether pour L3(2)

We prove that for a system of indeterminates (X_a) indiced by the P^2(2), the projective plane over F_2, there exists a 3-3 correspondance compatible with the incidence structures of P^2(2), such that (X_a) is one of the orbits of it. We give two applications of this construction :

1. for any sufficientely general polynomial P in k[X] over a field k of car. 0, such that its Galois group is a subgroup of L3(2) ((=L2(7)), there exists Q in k[X] such that the Galois group of P-TQ over k(T) is L3(2). This implies in particular the so-called “arithmetical lifting property” for L3(2) over k.
2. There exists a generic polynomial in 7 parameters for polynomials of degree 7 with Galois group L3(2). This is equivalent to the fact that the Noether’s problem for L3(2) acting over the seven points of P^2(2) has a positive answer.