I have attached a snapshot of an example I'm working through in Galois Theory.
I was wondering if anyone could explain to me how the roots $\eta_{2}$ and $\eta_{3}$ are determined? The text mentions that the starting polynomial was $f(x) = x^3 - 3x + 1$ was somehow inspired by the root $\eta_{1} = \zeta + \zeta^8$ where $\zeta = e^{2\pi i/9}$.
I was able to verify that $\eta_{1} = \zeta + \zeta^8$ is indeed a root of $f(x)$, but I don't follow the remaining analysis of the relationship between $\eta_{1}$ and the other two roots.
I get the bigger picture idea of what this example is trying to illuminate: that is, an example of where the splitting field for a cubic polynomial is a degree $3$ extension, as opposed to (the previous example was this case) a degree $6$ extension. My questions summarized:
1) How are $\eta_{2}$ and $\eta_{3}$ obtained?
2) How can I show that $\eta_{1}$ is real (or is there a typo with the $2\cdot\text{cos}(2\pi/9)$ form?