Failure of Luroth's theorem for transcendence degree 3 Can somebody give an example which shows the failure of Luroth's theorem for transcendence degree 3 over $\mathbb{C}$
 A: In geometric terms:    
A complex variety $V$ of dimension $n$ is rational if there exists a birational map $\mathbb P^n  --\to V $ or, equivalently, if its function field is purely transcendental i.e. there exists a field isomorphism $Rat (V)\cong \mathbb C(t_1,\cdots, t_n)$ .
More generally $V$ is called unirational if there exists a rational surjective map $\mathbb P^N  --\to V $ or, equivalently, if its function field is a subfield of a purely transcendental field i.e. there exists a field embedding $Rat (V) \subset  \mathbb C(t_1,\cdots, t_N)$ .   
The Lüroth problem  will thus be solved negatively if one can prove that there exists a unirational variety which is not rational.
For dimension $\geq 3$ the existence of such a variety was  proved in 1971 by three teams of mathematicians, using different ideas : Artin-Mumford, Clemens-Griffiths and Iskovskikh-Manin.  
In conclusion, it is not true  that an extension field $\mathbb C\subset K$ of transcendence degree three over $\mathbb C$ which is a subfield $K\subset \mathbb C(t_1,\cdots,t_N)$ of a purely transcendental extension  must necessarily be isomorphic to a purely transcendental extension $\mathbb C(u_1,u_2,u_3)$.
[For transcendence degrees one or two  however it is true]
