<s>[INST] Splitting field of $x^3-3x+1$ I know there is a trick in that you can use $\omega + \omega ^{-1}$ where $\omega$ is a 9th root of unity but I can't see how this relates. I know this question has already been asked on here but there was no solution given using the trick.
 [/INST] $x = z + \frac 1z\\
x^3 - 3x + 1 = z^3 + \frac 1{z^3} + 1 = \frac {1}{z^3} (z^3 + e^\frac {2\pi i}{3})(z^3 + e^\frac {-2\pi i}{3})\\
x^3 - 3x + 1 = (x + 2\cos \frac {2\pi}{9})(x + 2\cos \frac {4\pi}{9})(x + 2\cos \frac {8\pi}{9})$ 
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