Let $F \leqslant E$ be a field extension.
If $a, b \in E$ are algebraic over $F$ then $a+b$ and $ab$ are algebraic as well. There is an short proof of this using the extension $E(a,b)$:
$[E(a,b):E]$ is finite so all elements are algebraic otherwise powers non-algebraic would form an infinite linearly independent set. In particular $a+b \in E(a,b)$ and $ab \in E(a,b)$ are algebraic.
But this proof doesn't construct the actual polynomial. Is there a constructive proof or any reason for such a proof not to exist?