If I have two algebraic numbers $\alpha,\beta$ such that $A(\alpha) = 0$ and $B(\beta)=0$ where $A,B \in \mathbb{Q}[x]$ are the minimal polynomials of $\alpha$ and $\beta$ respectively. Knowing only the coefficients (and degree) of $A$ and $ B$ is there an algorithm for generating the minimal polynomial of $\alpha+\beta$?
I asked a professor and he recommended computing powers of $A(x) + B(x)$ and claimed there was a way to reduce it to get the appropriate polynomial however I'm not seeing his method.
