This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:

Let $p$ be an odd prime and let $a$ be a primitive element modulo $p$, i.e. a generator of the group of units of $\mathbb{Z}/p\mathbb{Z}$. For an integer $b$, let $[b]$ denote the unique integer in the interval $[-\frac{p-1}{2},\frac{p-1}{2}]$ and congruent with $b$ modulo $p$.
Consider the following polynomial with integral coefficients: $P(X)=b_0+b_1X+b_2X+\dots+b_{\frac{p-1}{2}} X^{\frac{p-1}{2}}$, where $b_i$ is 1 if $[a^i]$ is odd and $0$ otherwise. The question is whether $\gcd(P(X),X^{\frac{p-1}{2}}-1)=1$. I verified this for $p\le 39401$.