I have started reading field theory.
Let $E$ be an extension field of $F$ and let $\alpha,\beta\in E$.Suppose that $\alpha $ is transcendental over $F$ but algebraic over $F(\beta)$.
Show that $\beta $ is algebraic over $F(\alpha)$.
Since $\alpha $ is algebraic over $F(\beta)\implies \exists p(x)\neq 0$ such that $p(\alpha)=0$ .So $p(x)$ must be a polynomial over $F(\beta)$ and not over $F$.
But these facts are taking me nowhere near the solution.Any help will be appreciated.