Study and computation of a Hurwitz space and totally real -extensions of

Developing on works by Fried, Völklein, Matzat, Malle, Dèbes, Wewers, we give a method for computing a Hurwitz space and illustrate it on some example of number theoretic interest: we study and compute a family of degree 9 covers of with monodromy group and having four branch points. We deduce explicit regular -extensions of the rational function field with totally real fibers. This gives rise to totally real polynomials over with Galois group .