Let $p \in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d > m > 1$ and $\alpha_1, \ldots, \alpha_m$ be a (strict) subset of all the distinct roots of $p$. Can the value of $c_1\alpha_1 + \ldots + c_m\alpha_m$, where $c_1, \ldots, c_m$ are non-zero rationals, be rational? I would appreciate any suggestions.
I imagine the answer is negative, and it can be somehow proven by taking the splitting field of $p$ and applying the automorphisms to the sum above.
Yes, it's possible.
Let $p(x) = x^4-2$, which is irreducible of degree $4$ (irreducibility follows from Eistenstein's Criterion at $p=2$).
The sum of the two real roots, $\sqrt[4]{2}$ and $-\sqrt[4]{2}$, is $0$; so is the sum of the two nonreal roots, $i\sqrt[4]{2}$, and $-i\sqrt[4]{2}$. So we can take $m=2$, $\alpha_1\in\{\sqrt[4]{2},i\sqrt[4]{2}\}$, and $\alpha_2=-\alpha_1$, $c_1=c_2=1$. The sum is $0$, which of course is rational.