The Lüroth problem asks whether every field K with C ⊂ K ⊂ C(x1,...,xn) is of the form C(y1,...,yp). 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.
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