How to prove $L(\alpha)/K(\alpha)$ is algebraic given $L/K$ is algebraic?

Question

Let $L/K$ be a field extension and $K/F$ an algebraic extension. Take $\alpha \in L$. I would like to prove that $K(\alpha)/F(\alpha)$ is always algebraic. Any comments would be appreciated. Thank you!

Answer 1

Hint: If $L=K(\beta)$, then $K(\alpha)=F(\alpha)(\beta)$. Then the fact that $L/K$ is algebraic means that you can take $\beta$ to satisfy a certain type of equation. Now, use the definition of algebraic to show that the same type of equation is satisfied in the later extension.

Answer 2

Let $U\subseteq K(\alpha)$ be the subfield of the elements that are algebraic over $F(\alpha)$. Clearly $K\subseteq U$. Also, $\alpha\in U$ because it is a root of $T-\alpha\in F(\alpha)[T]$. Therefore $K(\alpha)\subseteq U$ as well.