Suppose you are given an algebraic field extension $L \supset K$ and $\alpha,\beta \in L$ with $f(X) = \text{minpoly}_K(\alpha)(X)=a_0+...+a_{m-1} X^{m-1}+X^m$ and $g(X)=\text{minpoly}_K(\beta)(X)=b_0+...+b_{n-1} X^{n-1}+X^n$.
Are the coefficients $c_i$ of $h(X)=\text{minpoly}_K(\alpha+\beta)(X)=c_0+...+c_{n-1} X^{n-1}+X^n$ polynomial expressions with integer coefficients in $(a_0,...,a_m,b_0,...,b_n)$, i. e. is $c_i \in \Bbb Z[a_0,...,a_m,b_0,...,b_n]$?
(My idea was to look at $P(X)=\prod_{1 \le i \le m,1 \le j \le n}(X-\alpha_i-\beta_j) \in \Bbb Z[X;a_0,...,a_m,b_0,...,b_n]$, where $f(X)=\prod_{1 \le i \le m}(X-\alpha_i)$ and $g(X)=\prod_{1 \le j \le n}(X-\beta_j)$. However $P$ might not be irreducible, and I don't know how to procede in that case.)