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Let $F$ be a field and let $F(X)$ be the field of rational functions with coefficients in $F$. Let $K$ be any field such that $F\subseteq K\subseteq F(X)$ and $K\neq F$. Prove that $[F(X):K]\lt\infty$.

Can I get an idea how to approach this problem?

Thank you.

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  • $\begingroup$ The magic words are “Luroth’s Theorem”. See the second answer here $\endgroup$ – Arturo Magidin Mar 24 at 2:41
  • $\begingroup$ @ArturoMagidin found a beautiful algebraic proof. Thanks. $\endgroup$ – ChakSayantan Mar 24 at 3:24
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The thing is to show that $X$ is algebraic over $K$. Then $K(X)=F(X)$ will be finite dimensional over $K$.

Since $K \neq F$, there is a rational function $f(X)=\frac{P(X)}{Q(X)} \in K$ that is not constant ($P$ and $Q$ are in $F[X]$). Then $X$ is a root of $f(X).Q(T)-P(T)$ which is a polynomial in $K[T]$ not equal to $0$ (you should check that).

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