Sub-Sum of Roots of Unity

Question

Let $\alpha$ be an algebraic integer of a cyclotomic field, and let $\theta_1, \theta_2, ..., \theta_n$ be roots of unity such that $$\sum_{i=1}^n \theta_i = 2\alpha.$$ Does there necessarily exists a sub-sum of the $\theta_i$ that equals $\alpha$?

Answer 1

No. Here we use the cyclotomic polynomial $\Phi_{105}$. $$ \begin{align} \Phi_{105}(x)&= x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} \\ &\qquad+ x^{36} + x^{35} + x^{34} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} \\ &\qquad- x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} \\ &\qquad+ x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1. \end{align} $$ Note that the coefficient of $x^{41}$ and $x^7$ are $-2$. Let $\zeta=e^{2\pi i /105}$. Then $\Phi_{105}(x)$ is the minimal polynomial for $\zeta$ over integers. This means that there are no polynomials with integer coefficients of degree smaller than $48$, that has $\zeta$ as a root.

Also, the polynomial has all nonzero coefficient other than $x^{41}$, $x^7$ are $\pm 1$. If $x$ is a root of unity, then so is $-x$.

By evaluating $\Phi_{105}$ at $\zeta$, we obtain a sum of $210$-th root of unity, that equals $2(\zeta^{41}+\zeta^7)$, which will be used as $2\alpha$ in your question.

If there is such sub-sum equals $\zeta^{41}+\zeta^7$ exists, then there should be another monic polynomial over integers of degree $\leq 48$, having $\zeta$ as a root. This is impossible.