A problem in extension fields

Question

Let $E$ be an extension field of a field $F$. (That is, let $F$ be a subfield of the field $E$.) Let $\alpha$, $\beta$ $\in $ $E$ such that $\alpha$ is transcendental over $F$ but algebraic over the simple extension $F(\beta)$ of $F$. Is $\beta$ algebraic or transcendental over $F$? I know that $\beta$ is algebraic over $F(\alpha)$.

Here $F(\beta)$ is effectively the field of rational functions in $\beta$ with co-efficients in $F$.

Answer 1

Hint: $[F(\alpha,\beta):F]=[F(\alpha,\beta):F(\beta)][F(\beta):F]$.

Answer 2

An algebraic extension of an algebraic extension amounts to an algebraic extension.