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

Question

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.

Answer 1

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).