The Lüroth problem asks whether every field K with C ⊂ K ⊂ C(x₁,...,x_n) is of the form C(y₁,...,y_p). In geometric terms, if an algebraic variety can be parametrized by rational functions, can one find a one-to-one such parametrization?
After a brief historical survey, I will recall the counter-examples found in the 70's; then I will describe a quite simple (and new) counter-example, and its application to the study of finite simple groups of birational automorphisms of P³.