This question comes from finite groups and their representations.
Let $q$ be a power of a prime $p$ and $1\leq a\leq p-1$. Whether $a$ is a square in $\mathbb{F}_p$ depends not on $p$ but only on the congruence of $p$ modulo $4a$. On the other hand, if $\zeta$ is a primitive $n$th root of unity in $\mathbb{F}_q$ then whether $n$ is a square or not depends only on whether $2n\mid (q-1)$, so again the congruence of $q$ modulo some integer.
Since integers are sums of roots of unity, namely $1$, I am looking at whether there is a generalization of the above situations. Namely, let $\zeta$ be a primitive $n$th root of unity in an algebraically closed field $k$ of characteristic $p$, let $I$ be a multiset of integers, and let $\alpha=\sum_{i\in I} \zeta^i$ be a sum of powers of $\zeta$. If $\alpha$ lies in $\mathbb{F}_q$, then we ask whether $\alpha$ is a square in $\mathbb{F}_q$.
So the question is:
Given $n$ and $I$, does there exist some integer $m$ such that, for any two prime powers $q$ and $q'$ with $q\equiv q'\bmod m$, $\alpha$ is a square in $\mathbb{F}_q$ if and only if $\alpha$ is a square in $\mathbb{F}_{q'}$?
In other words, does $\alpha$ being a square depend on the congruence of $q$ modulo some integer, rather than in some more complicated way?
I think the answer is 'yes', and it involves Galois theory of cyclotomic fields (namely the fact that the Galois groups are abelian), but I'm really not clued up about algebraic number theory.
(This question is a much simplified version of a previous question that I asked earlier today. I deleted that question and started anew because this version is so different to the previous one that an edit wouldn't really help.)