It is well known that the degree of the n-th cyclotomic polynomial is $\varphi(n)$, where $\varphi$ is the Euler totient function. I define the ${minimal}$ sum to be of the form
\begin{align} \xi_0 + \sum_{i=1 }^{k} \xi_i = 0 \end{align}
where $\xi_0$ an non-negative integer, $\xi_i$'s roots of unity of some order, and no subsum on the left sums to 0. If the sum were to have $\xi_n$, the primitive n-th root, as one of the terms, does the fact that the n-th cyclotomic polynomial have degree $\varphi(n)$ imply that this minimal sum has $\varphi(n)$ terms?