If a monic rational polynomial of degree $p-1$ has a $p$-th root of unity as a root, where $p$ is prime, does that make it the cyclotomic polynomial $x^{p-1}+…+1$?
I think this is the same as asking whether a rational polynomial could have a real common factor with the cyclotomic polynomial, such as $x^2+2Re(\zeta)x+1$ where $\zeta$ is a $p$-th root of unity. So we can ask what rational vectors is contained in
$span_\mathbb{R}\{(1,2Re(\zeta),1,0,…),(0,1,2Re(\zeta),1,0,…),…,(0,…,0,1,2Re(\zeta),1)\}$, but I'm not sure where to go from here.
Best Answer
If you also require the polynomial to be monic, then yes. This is because the cyclotomic polynomial of degree $p-1$ is the minimal polynomial over $\mathbb Q$ for all (nontrivial) $p$-th roots of unity.