This post does not deal with any particular problem, but instead asks if anyone knows any name, reference, or particular theorem associated with the following property:
Let $\alpha$ be an algebraic number. We say $\alpha$ is 'basic' (this is just what I've been calling it) if for any finite sum of algebraic numbers $\beta_i$ such that $$\beta_1+\beta_2+...\beta_n=\alpha,$$
there exists $1\leq i\leq n $ such that $\deg(\beta_i)\geq \deg(\alpha)$ (where the degree of an algebraic number is taken to be the degree of its minimal polynomial over the rationals).
A preliminary result I have proved already is if $\alpha$ is basic, then $r\alpha+s$ is basic for any $r,s\in\mathbb{Q}$.
Any known results or simply a term to research would be greatly appreciated.