I am recently reading some Galois Theory, and a question occurred to me: What are the intermediate fields of $K$ of $\mathbb C(x_1,\dots,x_n)$, where $n$ is an arbitrary integer?
I am aware of a certain Luroth's Theorem, which says that when $K$ is of transcendence degree 1, then $K=\mathbb{C}(w)$, where $w\in \mathbb{C}(x_1,\dots,x_n)$.
However, can much be said if we drop the restriction on the trdeg of $K$? For example, is $K$ also generated by $trdeg(K)$ elements? Is it even finitely generated?
It would be great if someone could recommend some references where I might get further information on the subject.