Let $L/K$ be a field extension. Let $\alpha\in L$ be algebraic over $K$, with minimal polynomial $P(X)$ over $K$. Now if $\beta$ is algebraic over $K(\alpha)$ with minimal polynomial $Q(X)$, then it must also be algebraic over $K$ since $K(\alpha, \beta)/K(\alpha)/K$ is a tower of finite extensions.
Is there a systematic way of finding the minimal polynomial of $\beta$ over $K$ using $P(X)$ an $Q(X)$?