Brad Cooper schrieb:   
   >   
   > 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?   
   >   
      
   In a bit more of detail ...   
      
   This works because both z^13 - 1 and SUBST(w^3 + w^2 - 4*w + 1, w, z   
   + z^5 + z^8 + z^12) possess the (irreducible) rational factor z^12   
   + z^11 + z^10 + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1,   
   as one easily verifies. I can't tell off-hand how to arrive at something   
   like SUBST(w^3 + w^2 - 4*w + 1, w, z + z^5 + z^8 + z^12) that shares a   
   mayor factor with a given z^13 - 1, but expect the theory of cyclotomic   
   polynomials to provide an answer.   
      
   Finding a polynomial with integer coefficients (like w^3 + w^2 - 4*w   
   + 1) whose roots include a given algebraic number (like 2*COS(2*pi/13)   
   - 2*COS(3*pi/13)) amounts to finding an integer relation, which feat can   
   be efficiently accomplished by the PSLQ algorithm for example.   
      
   Martin.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)