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I'm doing some exercises in the cyclotomic extension $\mathbb{Q}_7$. I have $\omega=e^{i2\pi /7}$, and I know $\omega+\omega^6=\omega+\omega^{-1}=2\cos(2\pi/7)$. My book says that then $\omega$ is a solution to a quadratic over $\mathbb{Q}(\cos 2\pi /7)$, but I don't see why. I only see a polynomial of degree 6. Where's the quadratic?

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    Well, perhaps if you multiplied both sides of $\omega + \omega^{-1} = 2 \cos \left( \frac{2\pi}{7} \right)$ by $\omega$...2011-07-14

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