🚀 go-pugleaf

                          CORRECTION
  
           PLEASE  SUBSTITUTE THIS FOR PREVIOUS  VERSION

Dear Mr. C.G 
 
  You asked me to give a proof of the Fundamental Theorem of
Algebra by e-mail.  I post only in TIME HAS INERTIA.  Here is my own
proof (I am almost sure that I have published it somewhere).
 

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

   PROOF. Assume on the contrary that  a + bz +...+ kz^n   with nonzero k, 
 
has no root, i.e., never vanishes.  Thus,  REWRITING THE POLYNOMIAL

IN DESCENDING POWERS    kz^n + ... + bz + a     it follows that the

quotient (since by our assumption never vanishes)

(1)    1 / (kz^n + ...+ bz + a)         
  
has a derivative at every point of the z-plane.
   
Hence, in particular,  

(2)    0   is an analytic point of  1/(kz^n + ... + bz + a),  and there-

fore the latter has a Laurent expansion in powers of  z  valid in the entire 
plane  (since we are over complex numbers - this is  not true over real 
numbers). 
  
  However, the Laurent expansion can be obtained by LONG DIVISION of (1).
  
Performing the long division of (1), we obtain

(3)   1/(kz^n + ...+ bz + a)  =  (1/k) z^(-n) + ........
 
But (3) shows that  0  is an essential isolated singularity of 
                   
                    1/(kz^n + ...+ bz + a)
 
contradicting (2).  Hence our assumption is false and the Theorem is
proved.

P.S.  Note that the proof will fail if instead of a polynomial we con-
      sider an entire function  (i.e., a power series, i.e., so to speak 
"an infinite polynomial"). The reason is that one cannot rewrite the
power series in descending order. Indeed, there are nonvanishing power
series such as the power series of  e^z.
     With best regards,                              Alexander ABIAN
           

P.S.  I typed the previous version and I thought that I pressed "i"
      to check the spelling and the mistakes.  Apparently I pressed
      "s" because I could no longer edit the text.
       Also, I know all the standard proofs e.g., by Liouville's Thm.











>-- 
>   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)
-- 
   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)