1. Examples in low degree -- 2. Nilpotent and solvable groups as Galois groups over Q -- 3. Hilbert's irreducibility theorem -- 4. Galois extensions of Q(T) : first examples -- 5. Galois extensions of Q(T) given by torsion on elliptic curves -- 6. Galois extensions of C(T) -- 7. Rigidity and rationality on finite groups -- 8. Construction of Galois extensions of Q(T) by the rigidity method -- 9. The form Tr(x[superscript 2]) and its applications -- 10. Appendix : the large sieve inequality.