Let $F\subset E$ be a field extension, $S\subset E$ and elements in $S$ are all algebraic over $F$. Prove $F(S)$ is an algebraic extension over $F$. And then all algbraic elements of $E$ form a field extension over $F$.
My try:
1. To prove $F(S)$ is an algebraic extension over $F$, we need to prove every elemnents in $F(S)$ is algebraic.
2. To prove every elemnents in $F(S)$ is algebraic. And since elements in $F(S)$ is of form of product of $\sum_{s\in S} f_s s,f_s\in F$, we need to prove each kind of product of $\sum_{s\in S} f_s s,f_s\in F$ is algebraic.
3. To prove each kind of product of $\sum_{s\in S} f_s s,f_s\in F$ is algebraic, we only need to prove for any two algebraic elements $\alpha,\beta$, $\alpha\cdot\beta$, $\alpha+\beta$ is algebraic over $F$.
I already know how to prove the sum and product are algebraic. Now I am not sure these steps could finish the proof. Please give a verification.