🚀 go-pugleaf

Dear Mr. C.G.                             11-11-92

   I have rigorously proved in my posting: 
       
       Message ID: <abian.721303213@pv343f.vincent.iastate.edu>
the Fundamental Theorem of Algebra, i.e.,

  THEOREM.  Every nonconstant polynomial over Complex numbers has a root.


  Now, you are asking (by e-mail) for the GIST of the proof, with less
cosmetics.  O.K.
  No cosmetics, except you have to know that the proof will not hold for
polynomials over reals, since:  In Complex Analysis it is true that if a 
function has a derivative then it has derivatives of all orders -this is not 
true in Real Analysis (see my abovementioned posting).  

  PROOF.  Assume on the contrary, and without loss of generality, let
     1 + z + z^2  have no root, i.e.,  1 + z + z^2  never vanishes.
Long divide  1  by  1+z+z^2  in two ways - in descending and ascending 
powers of z,  i.e., 

    1/ (1+z+z^2)  =   1 - z + z^3 - z^4 + z^6 - ...

    1/ (z^2+z+1)  =   1/(z^2) - 1/(z^3) + 1/(z^5) - 1/(z^6) + ... 
   
The two expressions on the right side of =  must be identical. But 
they are not.  Contradiction.  Hence our assumption is false and the 
Theorem is proved.
    But, I have given the rigorous proof in my abovementioned posting.  
    I hope this will satisfy your craving for the gist.
                                              Alexander  ABIAN             
 

-- 
   The tendency of maintaining the status-quo, Reaction to provocation and
                The tendency of maintaining again a status-quo.  
    TIME HAS INERTIA  and some energy is lost to move Time forward  
  E = mcc  (Einstein)    must be replaced by    E = m(0) exp(-At) (Abian)