Study and computation of a Hurwitz space and totally real PSL_2(F_8)-extensions of Q

International audienceDeveloping on works by Fried, V\"{o}lklein, Matzat, Malle, Débes, Wewers, we give a method for computing a Hurwitz space and illustrate it on some example of number theoristic interest: we study and compute a family of degree~$9$ covers of~$\PP^1_\CC$ with monodromy group~$PSL_2(\FF_8)$ and having four branch points. We deduce explicit regular~$PSL_2(\FF_8)$-extensions of the rational function field~$\QQ(\varphi)$ with totally real fibers. This gives rise to totally real polynomials over~$\QQ$ with Galois group~$PSL_2(\FF_8)$