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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.

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As for the more basic aspects of finite generation and transcendence bases you might want to take a look into Lang's Algebra: with the help of the material one can find there you can show

  • every subfield $K$ of $C(x_1,\ldots ,x_n)$ is finitely generated,
  • $K$ is generated by $\mathrm{trdeg}(K)+1$ elements.

If $K$ is generated by $\mathrm{trdeg}(K)$ elements, it is itself a rational function field. It is known that for $n>1$ this is not true for all subfields of $C(x_1,\ldots ,x_n)$ that are proper extensions of $C$.

A (certain) criterion for the rationality of $K$ can be found in

J.Ohm, Subfields of rational function fields, Archiv der Mathematik 42, No 2, 1984.

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Ok. Thanks for the help! – Michael Luo Nov 18 '11 at 14:36

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