Let $z:=\exp{(2πi/7)}$, and $X:=\{ \sum _{j=1}^6 a_j z^j|a_j=0,1\}$ Then, how many $x\in X$ satisfy $\mathbb{Q}(x)=\mathbb{Q}(z)$ ?
It is clear that $\mathbb{Q}(x) \subset \mathbb{Q}(z)$. If $p(t) \in \mathbb{Q}(t)$ s.t., $p(x)=z$, $\mathbb{Q}(x)=\mathbb{Q}(z)$
I know $z^n$($n=1,\dots ,6$) satisfy it since $\mathbb{Z}/7\mathbb{Z}$ is a field. But I don't know how to check another elements.