Let $\alpha, \beta$ be algebraic numbers over $\mathbb{Q}$. Which necessary and sufficient conditions are known such that $$ \mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta) \text{ ?} \tag{$\ast$}$$
I know this is a too general question, but I'd like to know which theorems/facts come to your mind when you want to prove/disprove $(\ast)$ (even in particular cases!).
I start. Let $\mathcal{B}$ a base of $\mathbb{Q}(\alpha, \beta)$ over $\mathbb{Q}$; if $[\mathbb{Q}(\alpha, \beta):\mathbb{Q}]=n$, then $(\ast)$ is true if and only if $$\det\left([1]_{\mathcal{B}}\mid[\alpha+\beta]_{\mathcal{B}}\mid\cdots\mid[(\alpha+\beta)^{n-1}]_{\mathcal{B}}\right)\neq 0$$ where $[\alpha+\beta]_{\mathcal{B}}$ is $\alpha+\beta$ written in base $\mathcal{B}$.