theorem A celebrated case is Lüroth's problem, which Jacob Lüroth solved in the nineteenth century. Lüroth's problem concerns subextensions L of K(X), the rational functions in the single indeterminate X. Any such field is either equal to K or is also rational, i.e. L = K(F) for some rational function F. In geometrical terms this states that a non-constant rational map from the projective line to a curve C can only occur when C also has genus 0. That fact can be read off geometrically from the Riemann–Hurwitz formula. Even though Lüroth's theorem is often thought as a non elementary result, several elementary short proofs have been discovered for long. These simple proofs use only the basics of field theory and Gauss's lemma for primitive polynomials (see e.g.). Unirationality A unirational variety V over a field K is one dominated by a rational variety, so that its function field K(V) lies in a pure transcendental field of finite type (which can be chosen to be of finite degree over K(V) if K is infinite). The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo's theorem implies that for complex surfaces unirational implies rational, because both are characterized by the vanishing of both the arithmetic genus and the second plurigenus. Zariski found some examples (Zariski surfaces) in characteristic p > 0 that are unirational but not rational. showed that a cubic three-fold is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used an intermediate Jacobian. showed that all non-singular quartic threefolds are irrational, though some of them are unirational. found some unirational 3-folds with non-trivial torsion in their third cohomology group, which implies that they are not rational. For any field K, János Kollár proved in 2000 that a smooth cubic hypersurface of dimension at least 2 is unirational if it has a point defined over K. This is an improvement of many classical results, beginning with the case of cubic surfaces (which are rational varieties over an algebraic closure). Other examples of varieties that are shown to be unirational are many cases of the moduli space of curves. Rationally connected variety A rationally connected variety (or uniruled variety) V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of