From: abian@iastate.edu (Alexander Abian)
Subject: Re: TIME HAS INERTIA.   Att:  PRATT  FUND. THM. OF ALGEBRA 
Keywords: CORRECTION        FUNDAMENTAL THM. OF ALGEBRA
Organization: Iowa State University, Ames IA
Date: Wed, 18 Nov 1992 07:27:28 GMT

In <abian.721303213@pv343f.vincent.iastate.edu> abian@iastate.edu (Alexander Abian) writes:


Mr. PRATT - here is my proof of FTA:

 
>  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
>           

> PS.  I know all the standard proofs e.g., by Liouville's Thm.
       But no proof is as gorgeous as the above.  It is a
       beauty !  See especially  the shorter version of this
       proof under   THE GIST OFMY FTA PRROF in TIME HAS INERTIA










>>-- 
>>   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)
-- 
   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)
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