How to tell if an algebraic number is an element of another algebraic field extension?

Question

Suppose $F$ is a field, $F \subset K$ is an extension, and $\alpha$, $\beta \in K$ are algebraic.

How do I tell, given the minimal polynomials $f(x)$, $g(x)$ of $\alpha$, $\beta$ respectively, if $\beta$ is an element of $F[\alpha]$ ?

For example, consider $F = \Bbb Q$, $K = \Bbb C$, and $\alpha$ such that $\alpha^3 + \alpha + 1 = 0$, $\beta = i$. How can I proceed ?