Let $z=e^\frac{2πi}{7}$ and let $x=z+z^2+z^4$.Then
$1.$ $x\in \Bbb Q$.
$2.$ $x\in \Bbb Q(\sqrt {D})$ for some $D \gt 0$.
$3.$ $x\in \Bbb Q(\sqrt {D})$ for some $D \lt 0$.
$4.$ $x\in i\Bbb R$.
Here $z$ is seventh root of unity.Hence $1+z+z^2+z^3+z^4+z^5+z^6=0$ hence
$z^2+z^4+z=-1-z^3-z^5-z^6$.$z^2$ and $z^4$ are conjugate of each other.hence their addition will be some real no.similarly for $z^3$ and $z^5$.But how to proceed further?