Let $K$ be a field, $\alpha\in\bar{K}$, and $L/K$ a finite extension. How can we determine whether $\alpha\in L$, preferably in as much generality as possible?

Of course, there may be special cases where this is easy, e.g. $K\subset\mathbb{R}$ and $\alpha\in\mathbb{C}\setminus\mathbb{R}$. Another trick, using the field trace, is the described in exercise 16 of Chapter 2 of Marcus's *Number Fields*, though as far as I can tell this only works for the special case of radicals (since their traces are always 0).

There's only two general approaches I can think of at the moment: checking whether $\deg(\alpha)\mid [L:K]$, though this may not suffice; or somehow finding the minimal polynomial of $\alpha$ over $L$ (including proving that it is irreducible), which will be of degree 1 iff $\alpha\in L$ (not entirely sure how one would do this).

I'm wondering because I recently had the following messy situation: $K$ is the splitting field of a cubic $g\in\mathbb{Q}(t)[x]$, having $[K:\mathbb{Q}(t)]=3$, and $f_1$, $f_2\in K[x]$ are two cubics with splitting fields, $L_1$ and $L_2$ respectively, having $[L_1:K]=[L_2:K]=3$, and I wanted to know whether $L_1=L_2$. It would suffice to show $f_1$ has a root in $L_2$ or vice versa (since $L_1=K($any root of $f_1)$ and $L_2=K$(any root of $f_2)$, which led me to my question. Ultimately (and I still want to double-check my answer), I found that $L_1=L_2$, but I depended heavily on the specific properties of my $f_1$, $f_2$, and $g$.