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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 $Q(X)$?
I'm sure there must be a better way, but a simple way to find it is:
Any polynomial expression in $\alpha,\beta$ can be expanded into a sum of terms of the form $c \alpha^i \beta^j$, with $c \in L$. We can then use the polynomial expressions $\alpha^n + a_1 \alpha^{n - 1} + \dots + a_n = 0 \Rightarrow \alpha^n = -a_1 \alpha^{n - 1} - \dots - a_n$ to rewrite it into a form with $i < \deg(P)$ and $j < \deg(Q)$. Using this, we can expand $1,\beta,\beta^2,\dots$ into linear combinations of elements from our finite set of $\alpha^i \beta^j$, and so after a number of iterations (< deg(P)deg(Q)), they must become linearly dependent. Standard linear algebra can then be used to find the coefficients, yielding a polynomial over $L$, which has $\beta$ as a zero.
