TITLE: Methods of showing an element is / is not in a field
QUESTION [5 upvotes]: 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 $g$.

REPLY [5 votes]: See section 5.4 "The Subfield Problem and Applications" in Henri Cohen's book "A Course in Computational Algebraic Number Theory". This is an excellent reference for computational algebraic number theory and the first place one should look when investigating such topics.