The only elements in $K(x)$ which are algebraic over $K$ are the constants.

Question

let $K\subset L\subset K(x)$ be a subfield. show that $L=K$ or $K(x)/L$ is a finite extension. deduce that the only elements in $K(x)$ which are algebraic over $K$ are the constants.

If $L=K$ then we are done, then suppose that $L\ne K$. Now, how to show that $K(x)/L$ is finite?

Answer 1

Let $L$ include a non-constant $\dfrac{p(x)}{q(x)}$. We can call this $t$ and ask whether $x$ is algebraic over $K(t)$. The answer is yes, because $t\cdot q(x)-p(x)=0$.

Since $x$ is algebraic over $K(t)$, it's algebraic over $L$. Hence $K(x)$ is a finite extension of $L$.