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I'm searching for an interesting example or application of Luroth's theorem in in field/galois theory. I don't have much knowledge of algebraic geometry so the applications there are a little beyond me. I'd love to find an example that shows what's powerful about the theorem without requiring that knowledge.

Thank you in advance!

The theorem for reference:

Let F be a field, and let F(x) be the field of rational functions over F. If K is an intermediate field between F and F(x) then K is isomorphic to F(x).

P.S. I was also wondering if anyone could help me state the luroth problem purely in terms of field theory?

decribed here: https://encyclopediaofmath.org/wiki/L%C3%BCroth_problem

GuyH
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    Some related reading. For me instances of Lüroth are often byproducts of questions asking to identify the fixed field of a finite group of automorphisms of a rational function field. It's nice to know in advance that a single generator can be found. I won't link to those for I am not really using Lüroth at all in the argument. – Jyrki Lahtonen Dec 16 '21 at 18:33
  • @JyrkiLahtonen thank you for this. when you say a single generator, do you mean a single generator for the fixed field, i.e. that the fixed field will be simple? or do you mean a single generator for the group of automorphisms? – GuyH Dec 16 '21 at 19:03
  • Single generator in the sense that if $K\subset F \subset K(x)$ then $F=K(t)$ for some $t$. In the cases I have considererd $F$ was the fixed field of a finite group $G$ of automorphisms of $K(x)$. In such a case it is easy to produce the required $t$ by the usual methods (Lüroth not needed). Of course, $K(x)/F$ need not be Galois in general. – Jyrki Lahtonen Dec 16 '21 at 19:39

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