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