# $K$ is intermediate field between $F$ and $F(x)$. then $\operatorname{dim}F(x)$ over $K$ is finite.

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.

• The magic words are “Luroth’s Theorem”. See the second answer here – Arturo Magidin Mar 24 at 2:41
• @ArturoMagidin found a beautiful algebraic proof. Thanks. – ChakSayantan Mar 24 at 3:24

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).