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# complex number + quadratic problem

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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 p and q.

Here's an interesting problem I've been stumped on, any pointers and help? Thanks!

Feb 24, 2020

#1
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Like so:

Feb 24, 2020
#3
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omega doesnt have to be ei*(2pi/7) it can be ei*(4pi/7)

Guest Feb 24, 2020
#4
+29252
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Correct.  I just went for the simplest option!

Alan  Feb 24, 2020
#2
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Hi Alan

Could you tell me how you get from the last but one line to the last one, (without the use of a calculator). ?

Feb 24, 2020
#5
+29252
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I cheated and used a calculator (well, Mathcad's calculation facilities).

Alan  Feb 24, 2020
#6
+29252
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Thinking about it further, I made the last few lines of the derivation of q more complicated than necessary.  Going back a few lines from the end we can do the following:

Alan  Feb 24, 2020