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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: $P(X)=b_0+b_1X+b_2X^2+\dots+b_{\frac{p-1}{2}} X^{\frac{p-1}{2}}$, 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 $p\le 39401$.

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  • $\begingroup$ I also verified using computers that for $p\le 1000$, $P(X)$ is of the form $X^k Q(X)$ with $Q(X)$ irreducible except for $p=223, 367, 727$ or $751$. In these four exceptions $P(X)=X^k Q(X) R(X)$, with $Q(X)$ and $R(X)$ irreducible and $Q$ either $X+1$ or $X^2+1$. Observe that in these exceptions $p\equiv -1 \mod 4$. It seems to me that there is some reason for this. $\endgroup$
    – Angel del Rio
    Commented Apr 8, 2015 at 12:06
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    $\begingroup$ The earlier post was mathoverflow.net/questions/202158/… $\endgroup$
    – Gerry Myerson
    Commented Apr 8, 2015 at 12:53
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    $\begingroup$ Another equivalent form of the question: Let $k\mod p$ denote the representative of $k$ modulo $p$ in the interval $[-n,n]$ where $n=\frac{p−1}{2}$. Consider the $n\times n$ matrix having $1$ in the $(i,j)$-th entry if $ij \mod p$ is odd and $0$ otherwise. Is A invertible in the ring of rational matrices? $\endgroup$
    – Angel del Rio
    Commented Apr 14, 2015 at 16:27

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