From: brad.cooper_17@bigpond.com   
      
   Hi,   
      
   I have a question from the book   
      
   The Theory and Use of the COMPLEX VARIABLE by S.L. GREEN (1939)   
      
   If a is a complex root of z^13 = 1, prove that   
      
   w = a + a^5 + a^8 + a^12   
      
    is a root of   
      
   z^3 + z^2 - 4*z + 1 = 0       (1)   
      
      
   It is easy to find w = 2*cos(2*PI/13) - 2*cos(3*PI/13) which is real.   
      
   If I substitute this value of w into (1) I verify the result.   
      
      
   I have been trying to figure out what led S.L. GREEN to discover equation (1)   
   in the first place. It is on the whole, unsatisfying to simply verify the   
   result.   
      
   When I type    2*cos(2*PI/13) - 2*cos(3*PI/13)    into Wolfram Alpha I am   
   astonished to find that Wolfram Alpha establishes equation (1) and then says   
   this is a root of equation (1).   
      
   Can someone please help me and explain what led both S.L. GREEN and Wolfram   
   Alpha to equation (1) without them knowing it in advance?   
      
   Regards,   
   Brad   
      
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