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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
avatar+29252 
+5

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
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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
avatar+29252 
+2

I cheated and used a calculator (well, Mathcad's calculation facilities).

Alan  Feb 24, 2020
 #6
avatar+29252 
+4

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

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