Here $\alpha$ is a root of $x^6+x^3+1$.
What I figured out:
let $\beta = e^{\frac{2\pi i}{9}}$. $\mathbb Q(\alpha)$ is the splitting field of $x^6+x^3+1$. Such an embedding must fix $\mathbb Q$, and send a root of $x^6+x^3+1$ to another root. So there are in total $6$ of them. How should I write them down explicitly?
Note that $$x^6 + x^3 + 1 = \frac{x^9 - 1}{x^3 - 1},$$ thus the roots of $x^6 + x^3 + 1$ are the $9$th roots of unity which are not $3$rd roods of unity, i.e. the primitive $9$th roots of unity. The number $\beta = e^{2\pi i/9}$ is a primitive $9$th root of unity, and every other primitive $9$th root of unity can be written as $\beta^m$ for some $1 \leq m \leq 9$ with $\gcd(9,m) = 1$.