For a rational prime $p>3$, denote by $\zeta$ a fixed primitive root of unity of degree $p$, and let ${\mathbb K}={\mathbb Q}(\zeta)$ be the $p$th cyclotomic field. Consider the set of all non-zero elements of $\mathbb K$ of the form $a_0+a_1\zeta+\dotsb+a_{p-2}\zeta^{p-2}$ with $a_0,\dotsc,a_{p-2}\in\{0,1\}$. There are finitely many such elements; hence, the set of all their (ideal) prime divisors is finite, too. Can anything intelligent be said about this set? Say, what are the rational primes lying below these ideal primes?
Thanks in advance for any insights!