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Let \(\omega \) be a complex number such that \(\omega^7 = 1\) and \(\omega \neq 1\). Let \(\alpha = \omega + \omega^2 + \omega^4\) and \(\beta = \omega^3 + \omega^5 + \omega^6\). Then \(\alpha\) and \(\beta\) are roots of the quadratic \(x^2 + px + q = 0\) for some integers p and q. Find the ordered pair (p,q).

 Sep 4, 2020
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(p,q) = (8,-5).

 Sep 4, 2020

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