| The L\"uroth problem asks whether every field $K$ with $\mathbb{C}\subset K \subset \mathbb{C}(x_1,\ldots ,x_n)$ is of the form $\mathbb{C}(y_1,\ldots ,y_p)$. 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. Finally I will explain its relation with the study of the finite groups of birational automorphisms of $\mathbb{P}^3$. |