From: hebisch@antispam.uni.wroc.pl   
      
   Brad Cooper  wrote:   
   > 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?   
      
   S.L. GREEN probably started with Galois theory.  Consider   
   subfield K of complex numbers generated by roots of z^13 - 1.   
   Mapping z -> z^k for k=1,...,12 on roots when extended linearly   
   over Q give you authomorphizm of K.  It follows that group G   
   of authomorphims of K is izomorphic to multiplicative subgroup   
   of Z_13.  Now, {1, 5, 8, 12} is a subgroup of Z_13.  If known   
   that if H is a subgroup of G, then elements x in K such that   
   hx = x for all h in H form a subfield L of K.  Natural way to   
   form elements of L is for fixed x take \sum hx where suo is   
   over all elements in H.  Taking x = a we get   
      
   a + a^5 + a^8 + a^12   
      
   that is w.  Galois theory says that w satisfies equation   
   of degree at most 3 over integers.  That means that   
   w^3 can be expressed as lineare combination with rational   
   coefficients of w^2, w, and 1.  In other words we have equation   
      
   c_2w^2 + c_1w + c_0 = w^3   
      
   It is also known that a^k, k=0,...,11 form basis of K over Q.   
   Expressiong w^l in this basis gives us system of 12   
   equations with 3 unknows from which we can determine c_2, c_1   
   and c_0.  Doing this whole calculation is somewhat tedious so   
   GREEN could have invented some shortcuts.  OTOH mathematicians   
   in the past did substantial calculations by hand, so quite   
   possible that GREEN just did calculations as outlined above.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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