Let $n$ be a positive integer, $k=0,\cdots,n-1$, $\omega_k=e^{\tfrac{2\pi i}{n}k}$ be the roots of unity, $c_k \in \mathcal{Z}$ be integer coefficients, trivial and non-trivial be two subcategories for $c_k$ defined as follows: Trivial is when $c_k$ contains equal integers that are evenly spaced under modulo addition of a composite divisor of $n$, and non-trivial $c_k$ is when evenly spaced integers under modulo addition of a composite divisor of $n$ are not equal. An example of a trivial case for $n=6$ is $$ (2)\omega_0+(2)\omega_1+(1)\omega_2+(2)\omega_3+(2)\omega_4+(1)\omega_5=0 $$ because $(2)\omega_0+(2)\omega_3=0$, and $(2)\omega_1+(2)\omega_4=0$, and $(1)\omega_2+(1)\omega_5=0$. Then regarding the linear independence of $\sum_k c_k \omega_k = 0$ consider the following: Does their exist a non-trivial case?
Note: Evenly spaced under modulo addition of a composite divisor of $n$ means the following: Let $m_p$ be a composite divisors of $n$ such that $n=m_0 m_1 \cdots$, then the trivial case of $c_k$ satisfies $c_k=c_j$ for $(k-j) \mod m_p =0$.
I am not sure I understood your definition correctly but I believe the fact that the $n$th cyclotomic field extension of $Q$ has degree $\phi(n)$ says that there are no non trivial combinations.