Let us say that an algebraic number $\alpha \in \mathbf{R}$ is non-positively algebraic if it is the root of a monic polynomial $p(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$ where all coefficients $a_{i} \in \mathbf{Q}$ satisfy $a_i \leq 0$. Is anything known about this class of algebraic numbers? In particular, is there some non-positive algebraic number $\alpha$ which is not non-positively algebraic? Of course, all negative rationals are non-positively algebraic.
Contrast this with positively algebraic numbers, which is when the condition on the $a_i$ is $a_i \geq 0$. This has been somewhat studied, see e.g. here. There are some algebraic numbers which are not positively algebraic. For example, the golden ratio $\varphi$ is non-positively algebraic but not positively algebraic.